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| Meta Title | Differential Equations - Laplace Transforms |
| Meta Description | In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms. |
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Section 4.2 : Laplace Transforms
As we saw in the last
section
computing Laplace transforms directly can be fairly complicated. Usually we just use a
table of transforms
when actually computing Laplace transforms. The table that is provided here is not an all-inclusive table but does include most of the commonly used Laplace transforms and most of the commonly needed formulas pertaining to Laplace transforms.
Before doing a couple of examples to illustrate the use of the table let’s get a quick fact out of the way.
Fact
Given
f
(
t
)
and
g
(
t
)
then,
L
{
a
f
(
t
)
+
b
g
(
t
)
}
=
a
F
(
s
)
+
b
G
(
s
)
for any constants
a
and
b
.
In other words, we don’t worry about constants and we don’t worry about sums or differences of functions in taking Laplace
transforms. All that we need to do is take the transform of the individual functions, then put any constants back in and add or subtract the results back up.
So, let’s do a couple of quick examples.
Example 1
Find the Laplace transforms of the given functions.
f
(
t
)
=
6
e
−
5
t
+
e
3
t
+
5
t
3
−
9
g
(
t
)
=
4
cos
(
4
t
)
−
9
sin
(
4
t
)
+
2
cos
(
10
t
)
h
(
t
)
=
3
sinh
(
2
t
)
+
3
sin
(
2
t
)
g
(
t
)
=
e
3
t
+
cos
(
6
t
)
−
e
3
t
cos
(
6
t
)
Show All Solutions
Hide All Solutions
Show Discussion
Okay, there’s not really a whole lot to do here other than go to the
table
, transform the individual functions up, put any constants back in and then add or subtract the results.
We’ll do these examples in a little more detail than is typically used since this is the first time we’re using the tables.
a
f
(
t
)
=
6
e
−
5
t
+
e
3
t
+
5
t
3
−
9
Show Solution
F
(
s
)
=
6
1
s
−
(
−
5
)
+
1
s
−
3
+
5
3
!
s
3
+
1
−
9
1
s
=
6
s
+
5
+
1
s
−
3
+
30
s
4
−
9
s
b
g
(
t
)
=
4
cos
(
4
t
)
−
9
sin
(
4
t
)
+
2
cos
(
10
t
)
Show Solution
G
(
s
)
=
4
s
s
2
+
(
4
)
2
−
9
4
s
2
+
(
4
)
2
+
2
s
s
2
+
(
10
)
2
=
4
s
s
2
+
16
−
36
s
2
+
16
+
2
s
s
2
+
100
c
h
(
t
)
=
3
sinh
(
2
t
)
+
3
sin
(
2
t
)
Show Solution
H
(
s
)
=
3
2
s
2
−
(
2
)
2
+
3
2
s
2
+
(
2
)
2
=
6
s
2
−
4
+
6
s
2
+
4
d
g
(
t
)
=
e
3
t
+
cos
(
6
t
)
−
e
3
t
cos
(
6
t
)
Show Solution
G
(
s
)
=
1
s
−
3
+
s
s
2
+
(
6
)
2
−
s
−
3
(
s
−
3
)
2
+
(
6
)
2
=
1
s
−
3
+
s
s
2
+
36
−
s
−
3
(
s
−
3
)
2
+
36
Make sure that you pay attention to the difference between a “normal” trig function and hyperbolic functions. The only difference between them is the “
+
a
2
” for the “normal” trig functions becomes a “
−
a
2
” in the hyperbolic function! It’s very easy to get in a hurry and not pay attention and grab the wrong formula. If you don’t recall the definition of the hyperbolic functions see the notes for the
table
.
Let’s do one final set of examples.
Example 2
Find the transform of each of the following functions.
f
(
t
)
=
t
cosh
(
3
t
)
h
(
t
)
=
t
2
sin
(
2
t
)
g
(
t
)
=
t
3
2
f
(
t
)
=
(
10
t
)
3
2
f
(
t
)
=
t
g
′
(
t
)
Show All Solutions
Hide All Solutions
a
f
(
t
)
=
t
cosh
(
3
t
)
Show Solution
This function is not in the table of Laplace transforms. However, we can use
#30
in the table to compute its transform. This will correspond to #30 if we take
n=1
.
F
(
s
)
=
L
{
t
g
(
t
)
}
=
−
G
′
(
s
)
,
where
g
(
t
)
=
cosh
(
3
t
)
So, we then have,
G
(
s
)
=
s
s
2
−
9
G
′
(
s
)
=
−
s
2
+
9
(
s
2
−
9
)
2
Using #30 we then have,
F
(
s
)
=
s
2
+
9
(
s
2
−
9
)
2
b
h
(
t
)
=
t
2
sin
(
2
t
)
Show Solution
This part will also use
#30
in the table. In fact, we could use #30 in one of two ways. We could use it with
n
=
1
.
H
(
s
)
=
L
{
t
f
(
t
)
}
=
−
F
′
(
s
)
,
where
f
(
t
)
=
t
sin
(
2
t
)
Or we could use it with
n
=
2
.
H
(
s
)
=
L
{
t
2
f
(
t
)
}
=
F
″
(
s
)
,
where
f
(
t
)
=
sin
(
2
t
)
Since it’s less work to do one derivative, let’s do it the first way. So, using
#9
we have,
F
(
s
)
=
4
s
(
s
2
+
4
)
2
F
′
(
s
)
=
−
12
s
2
−
16
(
s
2
+
4
)
3
The transform is then,
H
(
s
)
=
12
s
2
−
16
(
s
2
+
4
)
3
c
g
(
t
)
=
t
3
2
Show Solution
This part can be done using either
#6
(with
n
=
2
) or
#32
(along with
#5
). We will use #32 so we can see an example of this. In order to use #32 we’ll need to notice that
∫
0
t
v
d
v
=
2
3
t
3
2
⇒
t
3
2
=
3
2
∫
0
t
v
d
v
Now, using #5,
f
(
t
)
=
t
F
(
s
)
=
π
2
s
3
2
we get the following.
G
(
s
)
=
3
2
(
π
2
s
3
2
)
(
1
s
)
=
3
π
4
s
5
2
This is what we would have gotten had we used #6.
d
f
(
t
)
=
(
10
t
)
3
2
Show Solution
For this part we will use
#24
along with the answer from the previous part. To see this note that if
g
(
t
)
=
t
3
2
then
f
(
t
)
=
g
(
10
t
)
Therefore, the transform is.
F
(
s
)
=
1
10
G
(
s
10
)
=
1
10
(
3
π
4
(
s
10
)
5
2
)
=
10
3
2
3
π
4
s
5
2
e
f
(
t
)
=
t
g
′
(
t
)
Show Solution
This final part will again use
#30
from the table as well as
#35
.
L
{
t
g
′
(
t
)
}
=
−
d
d
s
L
{
g
′
}
=
−
d
d
s
{
s
G
(
s
)
−
g
(
0
)
}
=
−
(
G
(
s
)
+
s
G
′
(
s
)
−
0
)
=
−
G
(
s
)
−
s
G
′
(
s
)
Remember that
g
(
0
)
is just a constant so when we differentiate it we will get zero!
As this set of examples has shown us we can’t forget to use some of the general formulas in the
table
to derive new Laplace transforms for functions that aren’t explicitly listed in the table! |
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- [6\.4 Solving Logarithm Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveLogEqns.aspx)
- [6\.5 Applications](https://tutorial.math.lamar.edu/Classes/Alg/ExpLogApplications.aspx)
[Close submenu (7. Systems of Equations)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-2)[7\. Systems of Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-2)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Algebra](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-2)/7\. Systems of Equations
- [7\.1 Linear Systems with Two Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx)
- [7\.2 Linear Systems with Three Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsThreeVrble.aspx)
- [7\.3 Augmented Matrices](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrix.aspx)
- [7\.4 More on the Augmented Matrix](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx)
- [7\.5 Nonlinear Systems](https://tutorial.math.lamar.edu/Classes/Alg/NonlinearSystems.aspx)
[Close submenu (Calculus I)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/Calculus I
- [Open submenu (1. Review)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-11)[1\. Review](https://tutorial.math.lamar.edu/Classes/CalcI/ReviewIntro.aspx)
- [Open submenu (2. Limits)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-12)[2\. Limits](https://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx)
- [Open submenu (3. Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-13)[3\. Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx)
- [Open submenu (4. Applications of Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-14)[4\. Applications of Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivAppsIntro.aspx)
- [Open submenu (5. Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-15)[5\. Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx)
- [Open submenu (6. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-16)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntAppsIntro.aspx)
- [Open submenu (Appendix A. Extras)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-17)[Appendix A. Extras](https://tutorial.math.lamar.edu/Classes/CalcI/ExtrasIntro.aspx)
[Close submenu (1. Review)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[1\. Review](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/1\. Review
- [1\.1 Functions](https://tutorial.math.lamar.edu/Classes/CalcI/Functions.aspx)
- [1\.2 Inverse Functions](https://tutorial.math.lamar.edu/Classes/CalcI/InverseFunctions.aspx)
- [1\.3 Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx)
- [1\.4 Solving Trig Equations](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx)
- [1\.5 Trig Equations with Calculators, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcI.aspx)
- [1\.6 Trig Equations with Calculators, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcII.aspx)
- [1\.7 Exponential Functions](https://tutorial.math.lamar.edu/Classes/CalcI/ExpFunctions.aspx)
- [1\.8 Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/LogFcns.aspx)
- [1\.9 Exponential and Logarithm Equations](https://tutorial.math.lamar.edu/Classes/CalcI/ExpLogEqns.aspx)
- [1\.10 Common Graphs](https://tutorial.math.lamar.edu/Classes/CalcI/CommonGraphs.aspx)
[Close submenu (2. Limits)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[2\. Limits](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/2\. Limits
- [2\.1 Tangent Lines and Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/Tangents_Rates.aspx)
- [2\.2 The Limit](https://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx)
- [2\.3 One-Sided Limits](https://tutorial.math.lamar.edu/Classes/CalcI/OneSidedLimits.aspx)
- [2\.4 Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx)
- [2\.5 Computing Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx)
- [2\.6 Infinite Limits](https://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx)
- [2\.7 Limits At Infinity, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx)
- [2\.8 Limits At Infinity, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx)
- [2\.9 Continuity](https://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx)
- [2\.10 The Definition of the Limit](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx)
[Close submenu (3. Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[3\. Derivatives](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/3\. Derivatives
- [3\.1 The Definition of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx)
- [3\.2 Interpretation of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeInterp.aspx)
- [3\.3 Differentiation Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/DiffFormulas.aspx)
- [3\.4 Product and Quotient Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ProductQuotientRule.aspx)
- [3\.5 Derivatives of Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffTrigFcns.aspx)
- [3\.6 Derivatives of Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx)
- [3\.7 Derivatives of Inverse Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx)
- [3\.8 Derivatives of Hyperbolic Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffHyperFcns.aspx)
- [3\.9 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx)
- [3\.10 Implicit Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDIff.aspx)
- [3\.11 Related Rates](https://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx)
- [3\.12 Higher Order Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/HigherOrderDerivatives.aspx)
- [3\.13 Logarithmic Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx)
[Close submenu (4. Applications of Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[4\. Applications of Derivatives](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/4\. Applications of Derivatives
- [4\.1 Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/RateOfChange.aspx)
- [4\.2 Critical Points](https://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx)
- [4\.3 Minimum and Maximum Values](https://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx)
- [4\.4 Finding Absolute Extrema](https://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx)
- [4\.5 The Shape of a Graph, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtI.aspx)
- [4\.6 The Shape of a Graph, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx)
- [4\.7 The Mean Value Theorem](https://tutorial.math.lamar.edu/Classes/CalcI/MeanValueTheorem.aspx)
- [4\.8 Optimization](https://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx)
- [4\.9 More Optimization Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreOptimization.aspx)
- [4\.10 L'Hospital's Rule and Indeterminate Forms](https://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx)
- [4\.11 Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcI/LinearApproximations.aspx)
- [4\.12 Differentials](https://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx)
- [4\.13 Newton's Method](https://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx)
- [4\.14 Business Applications](https://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx)
[Close submenu (5. Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[5\. Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/5\. Integrals
- [5\.1 Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IndefiniteIntegrals.aspx)
- [5\.2 Computing Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingIndefiniteIntegrals.aspx)
- [5\.3 Substitution Rule for Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx)
- [5\.4 More Substitution Rule](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx)
- [5\.5 Area Problem](https://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx)
- [5\.6 Definition of the Definite Integral](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx)
- [5\.7 Computing Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx)
- [5\.8 Substitution Rule for Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx)
[Close submenu (6. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/6\. Applications of Integrals
- [6\.1 Average Function Value](https://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx)
- [6\.2 Area Between Curves](https://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx)
- [6\.3 Volumes of Solids of Revolution / Method of Rings](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx)
- [6\.4 Volumes of Solids of Revolution/Method of Cylinders](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx)
- [6\.5 More Volume Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx)
- [6\.6 Work](https://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx)
[Close submenu (Appendix A. Extras)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Appendix A. Extras](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-10)/Appendix A. Extras
- [A.1 Proof of Various Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx)
- [A.2 Proof of Various Derivative Properties](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx)
- [A.3 Proof of Trig Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx)
- [A.4 Proofs of Derivative Applications Facts](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx)
- [A.5 Proof of Various Integral Properties](https://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx)
- [A.6 Area and Volume Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/Area_Volume_Formulas.aspx)
- [A.7 Types of Infinity](https://tutorial.math.lamar.edu/Classes/CalcI/TypesOfInfinity.aspx)
- [A.8 Summation Notation](https://tutorial.math.lamar.edu/Classes/CalcI/SummationNotation.aspx)
- [A.9 Constant of Integration](https://tutorial.math.lamar.edu/Classes/CalcI/ConstantofIntegration.aspx)
[Close submenu (Calculus II)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/Calculus II
- [Open submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-19)[7\. Integration Techniques](https://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx)
- [Open submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-20)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/IntAppsIntro.aspx)
- [Open submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-21)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricIntro.aspx)
- [Open submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-22)[10\. Series & Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx)
- [Open submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-23)[11\. Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/VectorsIntro.aspx)
- [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-24)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcII/3DSpace.aspx)
[Close submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[7\. Integration Techniques](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/7\. Integration Techniques
- [7\.1 Integration by Parts](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx)
- [7\.2 Integrals Involving Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx)
- [7\.3 Trig Substitutions](https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx)
- [7\.4 Partial Fractions](https://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx)
- [7\.5 Integrals Involving Roots](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithRoots.aspx)
- [7\.6 Integrals Involving Quadratics](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithQuadratics.aspx)
- [7\.7 Integration Strategy](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationStrategy.aspx)
- [7\.8 Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx)
- [7\.9 Comparison Test for Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx)
- [7\.10 Approximating Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx)
[Close submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/8\. Applications of Integrals
- [8\.1 Arc Length](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx)
- [8\.2 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx)
- [8\.3 Center of Mass](https://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx)
- [8\.4 Hydrostatic Pressure](https://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx)
- [8\.5 Probability](https://tutorial.math.lamar.edu/Classes/CalcII/Probability.aspx)
[Close submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/9\. Parametric Equations and Polar Coordinates
- [9\.1 Parametric Equations and Curves](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx)
- [9\.2 Tangents with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx)
- [9\.3 Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx)
- [9\.4 Arc Length with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx)
- [9\.5 Surface Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaSurfaceArea.aspx)
- [9\.6 Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx)
- [9\.7 Tangents with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx)
- [9\.8 Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx)
- [9\.9 Arc Length with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx)
- [9\.10 Surface Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarSurfaceArea.aspx)
- [9\.11 Arc Length and Surface Area Revisited](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength_SurfaceArea.aspx)
[Close submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[10\. Series & Sequences](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/10\. Series & Sequences
- [10\.1 Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx)
- [10\.2 More on Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/MoreSequences.aspx)
- [10\.3 Series - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Basics.aspx)
- [10\.4 Convergence/Divergence of Series](https://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx)
- [10\.5 Special Series](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Special.aspx)
- [10\.6 Integral Test](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx)
- [10\.7 Comparison Test/Limit Comparison Test](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx)
- [10\.8 Alternating Series Test](https://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx)
- [10\.9 Absolute Convergence](https://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx)
- [10\.10 Ratio Test](https://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx)
- [10\.11 Root Test](https://tutorial.math.lamar.edu/Classes/CalcII/RootTest.aspx)
- [10\.12 Strategy for Series](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx)
- [10\.13 Estimating the Value of a Series](https://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx)
- [10\.14 Power Series](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx)
- [10\.15 Power Series and Functions](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeriesandFunctions.aspx)
- [10\.16 Taylor Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx)
- [10\.17 Applications of Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx)
- [10\.18 Binomial Series](https://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx)
[Close submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[11\. Vectors](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/11\. Vectors
- [11\.1 Vectors - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Vectors_Basics.aspx)
- [11\.2 Vector Arithmetic](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArithmetic.aspx)
- [11\.3 Dot Product](https://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx)
- [11\.4 Cross Product](https://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx)
[Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-18)/12\. 3-Dimensional Space
- [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcII/3DCoords.aspx)
- [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfLines.aspx)
- [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx)
- [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcII/QuadricSurfaces.aspx)
- [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcII/MultiVrbleFcns.aspx)
- [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFunctions.aspx)
- [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFcnsCalculus.aspx)
- [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx)
- [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArcLength.aspx)
- [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcII/Curvature.aspx)
- [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcII/Velocity_Acceleration.aspx)
- [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/CylindricalCoords.aspx)
- [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx)
[Close submenu (Calculus III)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/Calculus III
- [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-26)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcIII/3DSpace.aspx)
- [Open submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-27)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivsIntro.aspx)
- [Open submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-28)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivAppsIntro.aspx)
- [Open submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-29)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx)
- [Open submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-30)[16\. Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsIntro.aspx)
- [Open submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-31)[17\.Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegralsIntro.aspx)
[Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/12\. 3-Dimensional Space
- [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcIII/3DCoords.aspx)
- [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx)
- [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx)
- [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx)
- [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/MultiVrbleFcns.aspx)
- [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFunctions.aspx)
- [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFcnsCalculus.aspx)
- [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentNormalVectors.aspx)
- [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorArcLength.aspx)
- [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcIII/Curvature.aspx)
- [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcIII/Velocity_Acceleration.aspx)
- [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/CylindricalCoords.aspx)
- [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx)
[Close submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/13\. Partial Derivatives
- [13\.1 Limits](https://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx)
- [13\.2 Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx)
- [13\.3 Interpretations of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx)
- [13\.4 Higher Order Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/HighOrderPartialDerivs.aspx)
- [13\.5 Differentials](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx)
- [13\.6 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx)
- [13\.7 Directional Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx)
[Close submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/14\. Applications of Partial Derivatives
- [14\.1 Tangent Planes and Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx)
- [14\.2 Gradient Vector, Tangent Planes and Normal Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx)
- [14\.3 Relative Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx)
- [14\.4 Absolute Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/AbsoluteExtrema.aspx)
- [14\.5 Lagrange Multipliers](https://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx)
[Close submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/15\. Multiple Integrals
- [15\.1 Double Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx)
- [15\.2 Iterated Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx)
- [15\.3 Double Integrals over General Regions](https://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx)
- [15\.4 Double Integrals in Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/DIPolarCoords.aspx)
- [15\.5 Triple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx)
- [15\.6 Triple Integrals in Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TICylindricalCoords.aspx)
- [15\.7 Triple Integrals in Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx)
- [15\.8 Change of Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx)
- [15\.9 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx)
- [15\.10 Area and Volume Revisited](https://tutorial.math.lamar.edu/Classes/CalcIII/Area_Volume.aspx)
[Close submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[16\. Line Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/16\. Line Integrals
- [16\.1 Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx)
- [16\.2 Line Integrals - Part I](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx)
- [16\.3 Line Integrals - Part II](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx)
- [16\.4 Line Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields.aspx)
- [16\.5 Fundamental Theorem for Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)
- [16\.6 Conservative Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx)
- [16\.7 Green's Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx)
[Close submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[17\.Surface Integrals](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-25)/17\.Surface Integrals
- [17\.1 Curl and Divergence](https://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx)
- [17\.2 Parametric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx)
- [17\.3 Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx)
- [17\.4 Surface Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx)
- [17\.5 Stokes' Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx)
- [17\.6 Divergence Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx)
[Close submenu (Differential Equations)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/Differential Equations
- [Open submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-33)[1\. Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/IntroBasic.aspx)
- [Open submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-34)[2\. First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroFirstOrder.aspx)
- [Open submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-35)[3\. Second Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroSecondOrder.aspx)
- [Open submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-36)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx)
- [Open submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-37)[5\. Systems of DE's](https://tutorial.math.lamar.edu/Classes/DE/SystemsIntro.aspx)
- [Open submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-38)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/Classes/DE/SeriesIntro.aspx)
- [Open submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-39)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroHigherOrder.aspx)
- [Open submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-40)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/IntroBVP.aspx)
- [Open submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-41)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroPDE.aspx)
[Close submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[1\. Basic Concepts](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/1\. Basic Concepts
- [1\.1 Definitions](https://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx)
- [1\.2 Direction Fields](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx)
- [1\.3 Final Thoughts](https://tutorial.math.lamar.edu/Classes/DE/FinalThoughts.aspx)
[Close submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[2\. First Order DE's](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/2\. First Order DE's
- [2\.1 Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/Linear.aspx)
- [2\.2 Separable Equations](https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx)
- [2\.3 Exact Equations](https://tutorial.math.lamar.edu/Classes/DE/Exact.aspx)
- [2\.4 Bernoulli Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx)
- [2\.5 Substitutions](https://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx)
- [2\.6 Intervals of Validity](https://tutorial.math.lamar.edu/Classes/DE/IoV.aspx)
- [2\.7 Modeling with First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/Modeling.aspx)
- [2\.8 Equilibrium Solutions](https://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx)
- [2\.9 Euler's Method](https://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx)
[Close submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[3\. Second Order DE's](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/3\. Second Order DE's
- [3\.1 Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx)
- [3\.2 Real & Distinct Roots](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx)
- [3\.3 Complex Roots](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx)
- [3\.4 Repeated Roots](https://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx)
- [3\.5 Reduction of Order](https://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx)
- [3\.6 Fundamental Sets of Solutions](https://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx)
- [3\.7 More on the Wronskian](https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx)
- [3\.8 Nonhomogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx)
- [3\.9 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx)
- [3\.10 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx)
- [3\.11 Mechanical Vibrations](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx)
[Close submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/4\. Laplace Transforms
- [4\.1 The Definition](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx)
- [4\.2 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceTransforms.aspx)
- [4\.3 Inverse Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx)
- [4\.4 Step Functions](https://tutorial.math.lamar.edu/Classes/DE/StepFunctions.aspx)
- [4\.5 Solving IVP's with Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx)
- [4\.6 Nonconstant Coefficient IVP's](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx)
- [4\.7 IVP's With Step Functions](https://tutorial.math.lamar.edu/Classes/DE/IVPWithStepFunction.aspx)
- [4\.8 Dirac Delta Function](https://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx)
- [4\.9 Convolution Integrals](https://tutorial.math.lamar.edu/Classes/DE/ConvolutionIntegrals.aspx)
- [4\.10 Table Of Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx)
[Close submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[5\. Systems of DE's](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/5\. Systems of DE's
- [5\.1 Review : Systems of Equations](https://tutorial.math.lamar.edu/Classes/DE/LA_Systems.aspx)
- [5\.2 Review : Matrices & Vectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Matrix.aspx)
- [5\.3 Review : Eigenvalues & Eigenvectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx)
- [5\.4 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx)
- [5\.5 Solutions to Systems](https://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx)
- [5\.6 Phase Plane](https://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx)
- [5\.7 Real Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx)
- [5\.8 Complex Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx)
- [5\.9 Repeated Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx)
- [5\.10 Nonhomogeneous Systems](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx)
- [5\.11 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/SystemsLaplace.aspx)
- [5\.12 Modeling](https://tutorial.math.lamar.edu/Classes/DE/SystemsModeling.aspx)
[Close submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/6\. Series Solutions to DE's
- [6\.1 Review : Power Series](https://tutorial.math.lamar.edu/Classes/DE/PowerSeries.aspx)
- [6\.2 Review : Taylor Series](https://tutorial.math.lamar.edu/Classes/DE/TaylorSeries.aspx)
- [6\.3 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx)
- [6\.4 Euler Equations](https://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx)
[Close submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/7\. Higher Order Differential Equations
- [7\.1 Basic Concepts for *n*th Order Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/HOBasicConcepts.aspx)
- [7\.2 Linear Homogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOHomogeneousDE.aspx)
- [7\.3 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/HOUndeterminedCoeff.aspx)
- [7\.4 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx)
- [7\.5 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/HOLaplaceTransforms.aspx)
- [7\.6 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx)
- [7\.7 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/HOSeries.aspx)
[Close submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/8\. Boundary Value Problems & Fourier Series
- [8\.1 Boundary Value Problems](https://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx)
- [8\.2 Eigenvalues and Eigenfunctions](https://tutorial.math.lamar.edu/Classes/DE/BVPEvals.aspx)
- [8\.3 Periodic Functions & Orthogonal Functions](https://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx)
- [8\.4 Fourier Sine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSineSeries.aspx)
- [8\.5 Fourier Cosine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx)
- [8\.6 Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx)
- [8\.7 Convergence of Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/ConvergenceFourierSeries.aspx)
[Close submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32) [Pauls Notes](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/laplacetransforms.aspx#mm-32)/9\. Partial Differential Equations
- [9\.1 The Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx)
- [9\.2 The Wave Equation](https://tutorial.math.lamar.edu/Classes/DE/TheWaveEquation.aspx)
- [9\.3 Terminology](https://tutorial.math.lamar.edu/Classes/DE/PDETerminology.aspx)
- [9\.4 Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx)
- [9\.5 Solving the Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx)
- [9\.6 Heat Equation with Non-Zero Temperature Boundaries](https://tutorial.math.lamar.edu/Classes/DE/HeatEqnNonZero.aspx)
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### Section 4.2 : Laplace Transforms
As we saw in the last [section](https://tutorial.math.lamar.edu/classes/de/LaplaceDefinition.aspx) computing Laplace transforms directly can be fairly complicated. Usually we just use a [table of transforms](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) when actually computing Laplace transforms. The table that is provided here is not an all-inclusive table but does include most of the commonly used Laplace transforms and most of the commonly needed formulas pertaining to Laplace transforms.
Before doing a couple of examples to illustrate the use of the table let’s get a quick fact out of the way.
#### Fact
Given f(t) f ( t ) and g(t) g ( t ) then,
L{af(t)\+bg(t)}\=aF(s)\+bG(s)
L
{
a
f
(
t
)
\+
b
g
(
t
)
}
\=
a
F
(
s
)
\+
b
G
(
s
)
for any constants a a and b b.
In other words, we don’t worry about constants and we don’t worry about sums or differences of functions in taking Laplace transforms. All that we need to do is take the transform of the individual functions, then put any constants back in and add or subtract the results back up.
So, let’s do a couple of quick examples.
Example 1 Find the Laplace transforms of the given functions.
1. f(t)\=6e−5t\+e3t\+5t3−9
f
(
t
)
\=
6
e
−
5
t
\+
e
3
t
\+
5
t
3
−
9
2. g(t)\=4cos(4t)−9sin(4t)\+2cos(10t)
g
(
t
)
\=
4
cos
(
4
t
)
−
9
sin
(
4
t
)
\+
2
cos
(
10
t
)
3. h(t)\=3sinh(2t)\+3sin(2t)
h
(
t
)
\=
3
sinh
(
2
t
)
\+
3
sin
(
2
t
)
4. g(t)\=e3t\+cos(6t)−e3tcos(6t)
g
(
t
)
\=
e
3
t
\+
cos
(
6
t
)
−
e
3
t
cos
(
6
t
)
Show All Solutions Hide All Solutions
Show Discussion
Okay, there’s not really a whole lot to do here other than go to the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx), transform the individual functions up, put any constants back in and then add or subtract the results.
We’ll do these examples in a little more detail than is typically used since this is the first time we’re using the tables.
a
f(t)\=6e−5t\+e3t\+5t3−9
f
(
t
)
\=
6
e
−
5
t
\+
e
3
t
\+
5
t
3
−
9
Show Solution
F(s)\=61s−(−5)\+1s−3\+53\!s3\+1−91s\=6s\+5\+1s−3\+30s4−9s
F
(
s
)
\=
6
1
s
−
(
−
5
)
\+
1
s
−
3
\+
5
3
\!
s
3
\+
1
−
9
1
s
\=
6
s
\+
5
\+
1
s
−
3
\+
30
s
4
−
9
s
b
g(t)\=4cos(4t)−9sin(4t)\+2cos(10t)
g
(
t
)
\=
4
cos
(
4
t
)
−
9
sin
(
4
t
)
\+
2
cos
(
10
t
)
Show Solution
G(s)\=4ss2\+(4)2−94s2\+(4)2\+2ss2\+(10)2\=4ss2\+16−36s2\+16\+2ss2\+100
G
(
s
)
\=
4
s
s
2
\+
(
4
)
2
−
9
4
s
2
\+
(
4
)
2
\+
2
s
s
2
\+
(
10
)
2
\=
4
s
s
2
\+
16
−
36
s
2
\+
16
\+
2
s
s
2
\+
100
c
h(t)\=3sinh(2t)\+3sin(2t)
h
(
t
)
\=
3
sinh
(
2
t
)
\+
3
sin
(
2
t
)
Show Solution
H(s)\=32s2−(2)2\+32s2\+(2)2\=6s2−4\+6s2\+4
H
(
s
)
\=
3
2
s
2
−
(
2
)
2
\+
3
2
s
2
\+
(
2
)
2
\=
6
s
2
−
4
\+
6
s
2
\+
4
d
g(t)\=e3t\+cos(6t)−e3tcos(6t)
g
(
t
)
\=
e
3
t
\+
cos
(
6
t
)
−
e
3
t
cos
(
6
t
)
Show Solution
G(s)\=1s−3\+ss2\+(6)2−s−3(s−3)2\+(6)2\=1s−3\+ss2\+36−s−3(s−3)2\+36
G
(
s
)
\=
1
s
−
3
\+
s
s
2
\+
(
6
)
2
−
s
−
3
(
s
−
3
)
2
\+
(
6
)
2
\=
1
s
−
3
\+
s
s
2
\+
36
−
s
−
3
(
s
−
3
)
2
\+
36
Make sure that you pay attention to the difference between a “normal” trig function and hyperbolic functions. The only difference between them is the “\+a2 \+ a 2” for the “normal” trig functions becomes a “−a2 − a 2” in the hyperbolic function! It’s very easy to get in a hurry and not pay attention and grab the wrong formula. If you don’t recall the definition of the hyperbolic functions see the notes for the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#Note2).
Let’s do one final set of examples.
Example 2 Find the transform of each of the following functions.
1. f(t)\=tcosh(3t)
f
(
t
)
\=
t
cosh
(
3
t
)
2. h(t)\=t2sin(2t)
h
(
t
)
\=
t
2
sin
(
2
t
)
3. g(t)\=t32
g
(
t
)
\=
t
3
2
4. f(t)\=(10t)32
f
(
t
)
\=
(
10
t
)
3
2
5. f(t)\=tg′(t)
f
(
t
)
\=
t
g
′
(
t
)
Show All Solutions Hide All Solutions
a
f(t)\=tcosh(3t)
f
(
t
)
\=
t
cosh
(
3
t
)
Show Solution
This function is not in the table of Laplace transforms. However, we can use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) in the table to compute its transform. This will correspond to \#30 if we take *n=1*.
F(s)\=L{tg(t)}\=−G′(s),where g(t)\=cosh(3t)
F
(
s
)
\=
L
{
t
g
(
t
)
}
\=
−
G
′
(
s
)
,
where
g
(
t
)
\=
cosh
(
3
t
)
So, we then have,
G(s)\=ss2−9G′(s)\=−s2\+9(s2−9)2
G
(
s
)
\=
s
s
2
−
9
G
′
(
s
)
\=
−
s
2
\+
9
(
s
2
−
9
)
2
Using \#30 we then have,
F(s)\=s2\+9(s2−9)2
F
(
s
)
\=
s
2
\+
9
(
s
2
−
9
)
2
b
h(t)\=t2sin(2t)
h
(
t
)
\=
t
2
sin
(
2
t
)
Show Solution
This part will also use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) in the table. In fact, we could use \#30 in one of two ways. We could use it with n\=1 n \= 1*.*
H(s)\=L{tf(t)}\=−F′(s),where f(t)\=tsin(2t)
H
(
s
)
\=
L
{
t
f
(
t
)
}
\=
−
F
′
(
s
)
,
where
f
(
t
)
\=
t
sin
(
2
t
)
Or we could use it with n\=2 n \= 2.
H(s)\=L{t2f(t)}\=F′′(s),where f(t)\=sin(2t)
H
(
s
)
\=
L
{
t
2
f
(
t
)
}
\=
F
″
(
s
)
,
where
f
(
t
)
\=
sin
(
2
t
)
Since it’s less work to do one derivative, let’s do it the first way. So, using [\#9](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_9) we have,
F(s)\=4s(s2\+4)2F′(s)\=−12s2−16(s2\+4)3
F
(
s
)
\=
4
s
(
s
2
\+
4
)
2
F
′
(
s
)
\=
−
12
s
2
−
16
(
s
2
\+
4
)
3
The transform is then,
H(s)\=12s2−16(s2\+4)3
H
(
s
)
\=
12
s
2
−
16
(
s
2
\+
4
)
3
c
g(t)\=t32
g
(
t
)
\=
t
3
2
Show Solution
This part can be done using either [\#6](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_6) (with n\=2 n \= 2) or [\#32](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_32) (along with [\#5](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_5)). We will use \#32 so we can see an example of this. In order to use \#32 we’ll need to notice that
∫t0√vdv\=23t32⇒t32\=32∫t0√vdv
∫
0
t
v
d
v
\=
2
3
t
3
2
⇒
t
3
2
\=
3
2
∫
0
t
v
d
v
Now, using \#5,
f(t)\=√tF(s)\=√π2s32
f
(
t
)
\=
t
F
(
s
)
\=
π
2
s
3
2
we get the following.
G(s)\=32(√π2s32)(1s)\=3√π4s52
G
(
s
)
\=
3
2
(
π
2
s
3
2
)
(
1
s
)
\=
3
π
4
s
5
2
This is what we would have gotten had we used \#6.
d
f(t)\=(10t)32
f
(
t
)
\=
(
10
t
)
3
2
Show Solution
For this part we will use [\#24](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_24) along with the answer from the previous part. To see this note that if
g(t)\=t32
g
(
t
)
\=
t
3
2
then
f(t)\=g(10t)
f
(
t
)
\=
g
(
10
t
)
Therefore, the transform is.
F(s)\=110G(s10)\=110⎛⎜ ⎜ ⎜⎝3√π4(s10)52⎞⎟ ⎟ ⎟⎠\=10323√π4s52
F
(
s
)
\=
1
10
G
(
s
10
)
\=
1
10
(
3
π
4
(
s
10
)
5
2
)
\=
10
3
2
3
π
4
s
5
2
e
f(t)\=tg′(t)
f
(
t
)
\=
t
g
′
(
t
)
Show Solution
This final part will again use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) from the table as well as [\#35](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_35).
L{tg′(t)}\=−ddsL{g′}\=−dds{sG(s)−g(0)}\=−(G(s)\+sG′(s)−0)\=−G(s)−sG′(s)
L
{
t
g
′
(
t
)
}
\=
−
d
d
s
L
{
g
′
}
\=
−
d
d
s
{
s
G
(
s
)
−
g
(
0
)
}
\=
−
(
G
(
s
)
\+
s
G
′
(
s
)
−
0
)
\=
−
G
(
s
)
−
s
G
′
(
s
)
Remember that g(0) g ( 0 ) is just a constant so when we differentiate it we will get zero\!
As this set of examples has shown us we can’t forget to use some of the general formulas in the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) to derive new Laplace transforms for functions that aren’t explicitly listed in the table\!
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### Section 4.2 : Laplace Transforms
As we saw in the last [section](https://tutorial.math.lamar.edu/classes/de/LaplaceDefinition.aspx) computing Laplace transforms directly can be fairly complicated. Usually we just use a [table of transforms](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) when actually computing Laplace transforms. The table that is provided here is not an all-inclusive table but does include most of the commonly used Laplace transforms and most of the commonly needed formulas pertaining to Laplace transforms.
Before doing a couple of examples to illustrate the use of the table let’s get a quick fact out of the way.
#### Fact
Given f ( t ) and g ( t ) then,
L { a f ( t ) \+ b g ( t ) } \= a F ( s ) \+ b G ( s )
for any constants a and b.
In other words, we don’t worry about constants and we don’t worry about sums or differences of functions in taking Laplace transforms. All that we need to do is take the transform of the individual functions, then put any constants back in and add or subtract the results back up.
So, let’s do a couple of quick examples.
Example 1 Find the Laplace transforms of the given functions.
1. f
(
t
)
\=
6
e
−
5
t
\+
e
3
t
\+
5
t
3
−
9
2. g
(
t
)
\=
4
cos
(
4
t
)
−
9
sin
(
4
t
)
\+
2
cos
(
10
t
)
3. h
(
t
)
\=
3
sinh
(
2
t
)
\+
3
sin
(
2
t
)
4. g
(
t
)
\=
e
3
t
\+
cos
(
6
t
)
−
e
3
t
cos
(
6
t
)
Show All Solutions Hide All Solutions
Show Discussion
Okay, there’s not really a whole lot to do here other than go to the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx), transform the individual functions up, put any constants back in and then add or subtract the results.
We’ll do these examples in a little more detail than is typically used since this is the first time we’re using the tables.
a f ( t ) \= 6 e − 5 t \+ e 3 t \+ 5 t 3 − 9 Show Solution
F ( s ) \= 6 1 s − ( − 5 ) \+ 1 s − 3 \+ 5 3 \! s 3 \+ 1 − 9 1 s \= 6 s \+ 5 \+ 1 s − 3 \+ 30 s 4 − 9 s
b g ( t ) \= 4 cos ( 4 t ) − 9 sin ( 4 t ) \+ 2 cos ( 10 t ) Show Solution
G ( s ) \= 4 s s 2 \+ ( 4 ) 2 − 9 4 s 2 \+ ( 4 ) 2 \+ 2 s s 2 \+ ( 10 ) 2 \= 4 s s 2 \+ 16 − 36 s 2 \+ 16 \+ 2 s s 2 \+ 100
c h ( t ) \= 3 sinh ( 2 t ) \+ 3 sin ( 2 t ) Show Solution
H ( s ) \= 3 2 s 2 − ( 2 ) 2 \+ 3 2 s 2 \+ ( 2 ) 2 \= 6 s 2 − 4 \+ 6 s 2 \+ 4
d g ( t ) \= e 3 t \+ cos ( 6 t ) − e 3 t cos ( 6 t ) Show Solution
G ( s ) \= 1 s − 3 \+ s s 2 \+ ( 6 ) 2 − s − 3 ( s − 3 ) 2 \+ ( 6 ) 2 \= 1 s − 3 \+ s s 2 \+ 36 − s − 3 ( s − 3 ) 2 \+ 36
Make sure that you pay attention to the difference between a “normal” trig function and hyperbolic functions. The only difference between them is the “\+ a 2” for the “normal” trig functions becomes a “− a 2” in the hyperbolic function! It’s very easy to get in a hurry and not pay attention and grab the wrong formula. If you don’t recall the definition of the hyperbolic functions see the notes for the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#Note2).
Let’s do one final set of examples.
Example 2 Find the transform of each of the following functions.
1. f
(
t
)
\=
t
cosh
(
3
t
)
2. h
(
t
)
\=
t
2
sin
(
2
t
)
3. g
(
t
)
\=
t
3
2
4. f
(
t
)
\=
(
10
t
)
3
2
5. f
(
t
)
\=
t
g
′
(
t
)
Show All Solutions Hide All Solutions
a f ( t ) \= t cosh ( 3 t ) Show Solution
This function is not in the table of Laplace transforms. However, we can use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) in the table to compute its transform. This will correspond to \#30 if we take *n=1*.
F ( s ) \= L { t g ( t ) } \= − G ′ ( s ) , where g ( t ) \= cosh ( 3 t )
So, we then have,
G ( s ) \= s s 2 − 9 G ′ ( s ) \= − s 2 \+ 9 ( s 2 − 9 ) 2
Using \#30 we then have,
F ( s ) \= s 2 \+ 9 ( s 2 − 9 ) 2
b h ( t ) \= t 2 sin ( 2 t ) Show Solution
This part will also use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) in the table. In fact, we could use \#30 in one of two ways. We could use it with n \= 1*.*
H ( s ) \= L { t f ( t ) } \= − F ′ ( s ) , where f ( t ) \= t sin ( 2 t )
Or we could use it with n \= 2.
H ( s ) \= L { t 2 f ( t ) } \= F ″ ( s ) , where f ( t ) \= sin ( 2 t )
Since it’s less work to do one derivative, let’s do it the first way. So, using [\#9](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_9) we have,
F ( s ) \= 4 s ( s 2 \+ 4 ) 2 F ′ ( s ) \= − 12 s 2 − 16 ( s 2 \+ 4 ) 3
The transform is then,
H ( s ) \= 12 s 2 − 16 ( s 2 \+ 4 ) 3
c g ( t ) \= t 3 2 Show Solution
This part can be done using either [\#6](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_6) (with n \= 2) or [\#32](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_32) (along with [\#5](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_5)). We will use \#32 so we can see an example of this. In order to use \#32 we’ll need to notice that
∫ 0 t v d v \= 2 3 t 3 2 ⇒ t 3 2 \= 3 2 ∫ 0 t v d v
Now, using \#5,
f ( t ) \= t F ( s ) \= π 2 s 3 2
we get the following.
G ( s ) \= 3 2 ( π 2 s 3 2 ) ( 1 s ) \= 3 π 4 s 5 2
This is what we would have gotten had we used \#6.
d f ( t ) \= ( 10 t ) 3 2 Show Solution
For this part we will use [\#24](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_24) along with the answer from the previous part. To see this note that if
g ( t ) \= t 3 2
then
f ( t ) \= g ( 10 t )
Therefore, the transform is.
F ( s ) \= 1 10 G ( s 10 ) \= 1 10 ( 3 π 4 ( s 10 ) 5 2 ) \= 10 3 2 3 π 4 s 5 2
e f ( t ) \= t g ′ ( t ) Show Solution
This final part will again use [\#30](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_30) from the table as well as [\#35](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx#LT_35).
L { t g ′ ( t ) } \= − d d s L { g ′ } \= − d d s { s G ( s ) − g ( 0 ) } \= − ( G ( s ) \+ s G ′ ( s ) − 0 ) \= − G ( s ) − s G ′ ( s )
Remember that g ( 0 ) is just a constant so when we differentiate it we will get zero\!
As this set of examples has shown us we can’t forget to use some of the general formulas in the [table](https://tutorial.math.lamar.edu/classes/de/Laplace_Table.aspx) to derive new Laplace transforms for functions that aren’t explicitly listed in the table\! |
| Shard | 16 (laksa) |
| Root Hash | 6561424938415282416 |
| Unparsed URL | edu,lamar!math,tutorial,/classes/de/laplacetransforms.aspx s443 |