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[Explore Stack Internal](https://stackoverflow.co/internal/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams-compact-popover) # [Laplace transformations for dummies](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies) [Ask Question](https://math.stackexchange.com/questions/ask) Asked 15 years, 3 months ago Modified [8 years, 10 months ago](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies?lastactivity "2017-03-23 21:40:28Z") Viewed 69k times This question shows research effort; it is useful and clear 54 Save this question. Show activity on this post. Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical transformations. I searched the site and closest to an answer was [this](https://math.stackexchange.com/questions/954/inverse-of-laplace-transform). However, it is too complicated for me. - [intuition](https://math.stackexchange.com/questions/tagged/intuition "show questions tagged 'intuition'") - [integral-transforms](https://math.stackexchange.com/questions/tagged/integral-transforms "show questions tagged 'integral-transforms'") [Share](https://math.stackexchange.com/q/6661 "Short permalink to this question") Share a link to this question Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this question to receive notifications [edited Apr 13, 2017 at 12:19](https://math.stackexchange.com/posts/6661/revisions "show all edits to this post") [](https://math.stackexchange.com/users/-1/community) [Community](https://math.stackexchange.com/users/-1/community)Bot 1 asked Oct 13, 2010 at 4:49 [](https://math.stackexchange.com/users/1602/james-mertz) [James Mertz](https://math.stackexchange.com/users/1602/james-mertz) 1,22444 gold badges1313 silver badges2020 bronze badges 4 - 1 I would suggest you to go to books.google.com and read a bit here and a bit there - different presentations suit different people, some people learn by examples, other from theory and a third group of people learn from applications in exercises. AD - Stop Putin - – [AD - Stop Putin -](https://math.stackexchange.com/users/1154/ad-stop-putin "11,260 reputation") 2010-10-13 05:00:03 +00:00 [Commented Oct 13, 2010 at 5:00](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14143_6661) - What book are you using? That might allow people to get a gauge of what alternatives would suit you best. user1736 – [user1736](https://math.stackexchange.com/users/1736/user1736 "9,103 reputation") 2010-10-13 05:15:02 +00:00 [Commented Oct 13, 2010 at 5:15](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14144_6661) - Fundamentals of differential equations 7th Ed by Nagle, Saff, and Snider James Mertz – [James Mertz](https://math.stackexchange.com/users/1602/james-mertz "1,224 reputation") 2010-10-13 05:22:06 +00:00 [Commented Oct 13, 2010 at 5:22](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14145_6661) - This is not a bad book. Agustí Roig – [Agustí Roig](https://math.stackexchange.com/users/664/agust%C3%AD-roig "18,677 reputation") 2010-10-13 05:37:57 +00:00 [Commented Oct 13, 2010 at 5:37](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14147_6661) [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid answering questions in comments.") \| ## 7 Answers 7 Sorted by: [Reset to default](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies?answertab=scoredesc#tab-top) This answer is useful 36 Save this answer. Show activity on this post. There are beautiful video lessons at [MIT Opencourseware](http://ocw.mit.edu/index.htm). I'm particularly in love with [this](http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-19-introduction-to-the-laplace-transform/) presentation of the Laplace transform. [Share](https://math.stackexchange.com/a/6662 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this answer to receive notifications [edited Aug 11, 2014 at 19:55](https://math.stackexchange.com/posts/6662/revisions "show all edits to this post") [](https://math.stackexchange.com/users/130/i-j-kennedy) [I. J. Kennedy](https://math.stackexchange.com/users/130/i-j-kennedy) 4,17844 gold badges2929 silver badges4242 bronze badges answered Oct 13, 2010 at 4:55 [](https://math.stackexchange.com/users/664/agust%C3%AD-roig) [Agustí Roig](https://math.stackexchange.com/users/664/agust%C3%AD-roig) 18\.7k33 gold badges5252 silver badges9090 bronze badges 3 - 5 Very interesting relationship between the Laplace transform and power series, but it's not really clear why something like this is needed, or useful, for solving differential equations. dannum – [dannum](https://math.stackexchange.com/users/152081/dannum "2,639 reputation") 2017-04-17 15:33:43 +00:00 [Commented Apr 17, 2017 at 15:33](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment4602764_6662) - @danielson Maybe we should attend the next lesson: [ocw.mit.edu/courses/mathematics/…](https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-20-derivative-formulas/) Agustí Roig – [Agustí Roig](https://math.stackexchange.com/users/664/agust%C3%AD-roig "18,677 reputation") 2017-04-18 16:35:47 +00:00 [Commented Apr 18, 2017 at 16:35](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment4606510_6662) - 1 The lesson you pointed out talks about the Laplace transform as the continuous analog of ∑n∈Na(n)xn ∑ n ∈ N a ( n ) x n where n→t∈R n → t ∈ R and ∑n\=0∞a(n)xn→∫∞0f(t)e−stdt ∑ n \= 0 ∞ a ( n ) x n → ∫ 0 ∞ f ( t ) e − s t d t where s\=−ln(x) s \= − ln ⁡ ( x ) and f(t)\=a(et) f ( t ) \= a ( e t ) Good one\! Dr Potato – [Dr Potato](https://math.stackexchange.com/users/622884/dr-potato "1,033 reputation") 2024-11-21 04:40:50 +00:00 [Commented Nov 21, 2024 at 4:40](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment10720746_6662) [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 29 Save this answer. Show activity on this post. I have been teaching the Laplace Transform to a night degree class (mature) of civil engineers. They are good students but not great mathematicians. They couldn't follow the method of how we use the Laplace Transform to solve differential equations until I told them this story: > Suppose that you come across a poem written in English of whose meaning you don't understand. However suppose that you know a French-speaking gentleman who is a master of interpreting poems. So you translate the poem into French and send it to the French gentleman. The French gentleman writes a perfectly good interpretation of the poem in French and sends this back to you where you translate it back into English and you have the meaning of the poem. Obviously these are simple difficulties that these students are having but I still think it's a nice story. D'accord: Poem in English = Differential Equation. Interpretation in English = Solution of Differential Equation. Translate to French = Take Laplace Transform. Poem in French (better interpreter) = Algebraic Equation (easier to solve). Interpretation in French = Laplace Transform of Solution of Differential Equation. Translate back into English = Inverse Laplace Transform [Share](https://math.stackexchange.com/a/219688 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this answer to receive notifications [edited Mar 23, 2017 at 21:40](https://math.stackexchange.com/posts/219688/revisions "show all edits to this post") answered Oct 23, 2012 at 22:02 [](https://math.stackexchange.com/users/19352/jp-mccarthy) [JP McCarthy](https://math.stackexchange.com/users/19352/jp-mccarthy) 8,12411 gold badge3939 silver badges5858 bronze badges 0 [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 18 Save this answer. Show activity on this post. I'm going to come at this one from left-field. In quantum mechanics, we deal with infinite dimensional vector spaces (Hilbert spaces), so I tend to think of integral transforms in those terms. For instance, ∫∞−∞K(x,y)f(y)dy\=F(x) ∫ − ∞ ∞ K ( x , y ) f ( y ) d y \= F ( x ) can be thought of as Kf\=F K f \= F and x x and y y from the first equation are the indices of the infinite dimensional vectors and matrix (kernel) f f, F F, and K K. Using that interpretation, if K K is unitary then the integral is just a changing the bases of the function (Hilbert) space. In other words, the integral can be viewed as the decomposition of original vector, f f, in terms the new basis. For Fourier transforms the kernel is unitary, and while not true of Laplace transforms, the idea of it being a change of basis still holds. It should be noted that unlike in the finite case, in the infinite dimensional case care must be taken to ensure that the transform actually converges, but that is another problem entirely. [Share](https://math.stackexchange.com/a/6689 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 2.5](https://creativecommons.org/licenses/by-sa/2.5/ "The current license for this post: CC BY-SA 2.5") Cite Follow Follow this answer to receive notifications [edited Oct 13, 2010 at 13:53](https://math.stackexchange.com/posts/6689/revisions "show all edits to this post") answered Oct 13, 2010 at 13:13 [](https://math.stackexchange.com/users/2283/rcollyer) [rcollyer](https://math.stackexchange.com/users/2283/rcollyer) 1,96711 gold badge1717 silver badges2929 bronze badges 3 - 1 The Laplace transform certainly is not unitary. Robin Chapman – [Robin Chapman](https://math.stackexchange.com/users/226/robin-chapman "23,040 reputation") 2010-10-13 13:32:33 +00:00 [Commented Oct 13, 2010 at 13:32](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14199_6689) - You're right, in this case K−1≠K† K − 1 ≠ K † . Edited my answer to reflect this. rcollyer – [rcollyer](https://math.stackexchange.com/users/2283/rcollyer "1,967 reputation") 2010-10-13 13:49:31 +00:00 [Commented Oct 13, 2010 at 13:49](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment14203_6689) - @rcollyer Thanks for this more theoretical approach to the question. Would you be able to suggest a reference that discusses (ideally, rigorously) Laplace methods as a change of basis? Dragonsheep – [Dragonsheep](https://math.stackexchange.com/users/352344/dragonsheep "393 reputation") 2017-11-29 02:58:17 +00:00 [Commented Nov 29, 2017 at 2:58](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies#comment5248937_6689) [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 6 Save this answer. Show activity on this post. Refer <http://www.dspguide.com/CH32.PDF> for an excellent explanation of Laplace Transforms in the Electrical Domain. [Share](https://math.stackexchange.com/a/600022 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this answer to receive notifications answered Dec 9, 2013 at 16:22 [](https://math.stackexchange.com/users/114646/nereusf) [NereusF](https://math.stackexchange.com/users/114646/nereusf) 6111 silver badge11 bronze badge [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 3 Save this answer. Show activity on this post. Have also a look here - many great resources for the Laplace transform: <https://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition/2141#2141> [Share](https://math.stackexchange.com/a/7280 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 2.5](https://creativecommons.org/licenses/by-sa/2.5/ "The current license for this post: CC BY-SA 2.5") Cite Follow Follow this answer to receive notifications [edited Apr 13, 2017 at 12:58](https://math.stackexchange.com/posts/7280/revisions "show all edits to this post") [](https://math.stackexchange.com/users/-1/community) [Community](https://math.stackexchange.com/users/-1/community)Bot 1 answered Oct 20, 2010 at 10:03 [](https://math.stackexchange.com/users/346/vonjd) [vonjd](https://math.stackexchange.com/users/346/vonjd) 9,1201212 gold badges5656 silver badges8080 bronze badges [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 1 Save this answer. Show activity on this post. A Laplace transform is useful for turning (constant coefficient) ordinary differential equations into algebraic equations, and partial differential equations into ordinary differential equations (though I rarely see these daisy chained together). Let's say that you have an ordinary DE of the form ay′′(t)\+by′(t)\+cy(t)\=f(t)t\>0 a y ″ ( t ) \+ b y ′ ( t ) \+ c y ( t ) \= f ( t ) t \> 0 y(0)\=y0 y ( 0 ) \= y 0 y′(0)\=p0 y ′ ( 0 ) \= p 0 Then the above equation becomes (as2\+bs\+c)y^(s)−\[ay0s\+(ap0\+by0)\]\=f^(s) ( a s 2 \+ b s \+ c ) y ^ ( s ) − \[ a y 0 s \+ ( a p 0 \+ b y 0 ) \] \= f ^ ( s ) where y^ y ^ and f^ f ^ are Laplace transforms of y y and f f, respectively. Note that we have converted the ODE into an algebraic equation in which we solve for y^(s) y ^ ( s ). We find y(t) y ( t ) by inverse Laplace transformation, which is usually accomplished through tables, or contour integration if there is facility with that. Note that the initial conditions are built right into the equation we solve. There are numerous examples for using Laplace transforms in PDE's. [Here is a case](https://math.stackexchange.com/questions/480235/inverse-laplace-transform-of-bar-p-d-frack-0-sqrts-r-dsk-0-sqrts/481946#481946) I did in which I used LT's to solve the heat equation in two dimensions. [Share](https://math.stackexchange.com/a/531372 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this answer to receive notifications [edited Apr 13, 2017 at 12:21](https://math.stackexchange.com/posts/531372/revisions "show all edits to this post") [](https://math.stackexchange.com/users/-1/community) [Community](https://math.stackexchange.com/users/-1/community)Bot 1 answered Oct 18, 2013 at 19:26 [](https://math.stackexchange.com/users/53268/ron-gordon) [Ron Gordon](https://math.stackexchange.com/users/53268/ron-gordon) 142k1616 gold badges199199 silver badges324324 bronze badges [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful \-4 Save this answer. Show activity on this post. Think that the laplace transformation is a kind of a machine, the machine eats function of t f(t) out comes F(s). you do a transformation from time to frequency. Inside the machine you have this integral expression that you already know. it is similar when you transform from one vector space to another. for instance you go from R to R^2 [Share](https://math.stackexchange.com/a/531318 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite Follow Follow this answer to receive notifications answered Oct 18, 2013 at 18:36 [](https://math.stackexchange.com/users/101741/ronny) [Ronny](https://math.stackexchange.com/users/101741/ronny) 1 [Add a comment](https://math.stackexchange.com/questions/6661/laplace-transformations-for-dummies "Use comments to ask for more information or suggest improvements. 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