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Calculated Shard: 77 (from laksa014)

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curl -X POST \
  'http://laksa077.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://medium.com/@kailashnagarajan/laplace-transforms-an-untold-story-c08a27cc9012\')), getAhrefsUnparsedNoserviceFromURL(\'https://medium.com/@kailashnagarajan/laplace-transforms-an-untold-story-c08a27cc9012\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/medium.com\/@kailashnagarajan\/laplace-transforms-an-untold-story-c08a27cc9012","crawl_time":1774121988,"first_indexed_time":0,"http_code":200,"src_unparsed":"com,medium!\/@kailashnagarajan\/laplace-transforms-an-untold-story-c08a27cc9012 s443","src_root_hash":"13179037029838926277","history_drop_reason":null,"meta_title":"Laplace Transforms: An Untold Story | by Kailash Nagarajan | Medium","meta_descriptions":["Laplace Transforms: An Untold Story One of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way …"],"attrs_boilerpipe_text":"3 min read\nNov 10, 2018\n--\nOne of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into.\nBefore actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains?\nA function is anything that takes in an input and gives an output.Let’s consider a function f(x).\nPress enter or click to view image in full size\nSo for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this.\nA plot for f(x) = x²\nSimilarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc.\nNow, what is the definition of the domain of a given function? The\ndomain\nis the set of all possible x-values which will make the\nfunction\n“work” and will output real f(x)-values.\nOne such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’.\nLaplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’.\nHow do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain.\nThe formula for Laplace Transform\nF(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain.\nHere are the Laplace transforms of some basic functions.\nTable of Laplace Transforms of some basic functions.\nUsing Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain.\nOne of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations.\nFig1: Simple 1st order ODE\nThe above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below.\nPress enter or click to view image in full size\nLaplace transform of an nth order ODE\nThis amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations.\nIf you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function\nThank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from\nhere\n.","attrs_markdown":"[Sitemap](https:\/\/medium.com\/sitemap\/sitemap.xml)\n\n[Open in app](https:\/\/play.google.com\/store\/apps\/details?id=com.medium.reader&referrer=utm_source%3DmobileNavBar&source=post_page---top_nav_layout_nav-----------------------------------------)\n\nSign up\n\n[Sign in](https:\/\/medium.com\/m\/signin?operation=login&redirect=https%3A%2F%2Fmedium.com%2F%40kailashnagarajan%2Flaplace-transforms-an-untold-story-c08a27cc9012&source=post_page---top_nav_layout_nav-----------------------global_nav------------------)\n\n[Medium Logo](https:\/\/medium.com\/?source=post_page---top_nav_layout_nav-----------------------------------------)\n\nGet app\n\n[Write](https:\/\/medium.com\/m\/signin?operation=register&redirect=https%3A%2F%2Fmedium.com%2Fnew-story&source=---top_nav_layout_nav-----------------------new_post_topnav------------------)\n\n[Search](https:\/\/medium.com\/search?source=post_page---top_nav_layout_nav-----------------------------------------)\n\nSign up\n\n[Sign in](https:\/\/medium.com\/m\/signin?operation=login&redirect=https%3A%2F%2Fmedium.com%2F%40kailashnagarajan%2Flaplace-transforms-an-untold-story-c08a27cc9012&source=post_page---top_nav_layout_nav-----------------------global_nav------------------)\n\n![](https:\/\/miro.medium.com\/v2\/resize:fill:64:64\/1*dmbNkD5D-u45r44go_cf0g.png)\n\n# Laplace Transforms: An Untold Story\n[![Kailash Nagarajan](https:\/\/miro.medium.com\/v2\/resize:fill:64:64\/1*JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https:\/\/medium.com\/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------)\n\n[Kailash Nagarajan](https:\/\/medium.com\/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------)\n\n3 min read\n\n·\n\nNov 10, 2018\n\n\\--\n\nListen\n\nShare\n\nOne of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into.\n\nBefore actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains?\n\nA function is anything that takes in an input and gives an output.Let’s consider a function f(x).\n\nPress enter or click to view image in full size\n\n![]()\n\nSo for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this.\n\n![]()\n\nA plot for f(x) = x²\n\nSimilarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc.\n\nNow, what is the definition of the domain of a given function? The **domain** is the set of all possible x-values which will make the **function** “work” and will output real f(x)-values.\n\nOne such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’.\n\n### Laplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’.\nHow do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain.\n\n![]()\n\nThe formula for Laplace Transform\n\nF(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain.\n\nHere are the Laplace transforms of some basic functions.\n\n![]()\n\nTable of Laplace Transforms of some basic functions.\n### Using Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain.\nOne of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations.\n\n![]()\n\nFig1: Simple 1st order ODE\n\nThe above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below.\n\nPress enter or click to view image in full size\n\n![]()\n\nLaplace transform of an nth order ODE\n\nThis amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations.\n\nIf you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function\n\nThank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from [here](https:\/\/www.youtube.com\/watch?v=OiNh2DswFt4&list=PL6D8DCFEAF1A468DD).\n\n[Mathematics](https:\/\/medium.com\/tag\/mathematics?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Laplace Transforms](https:\/\/medium.com\/tag\/laplace-transforms?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Calculus](https:\/\/medium.com\/tag\/calculus?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Ode](https:\/\/medium.com\/tag\/ode?source=post_page-----c08a27cc9012---------------------------------------)\n\n\\--\n\n\\--\n\n[![Kailash Nagarajan](https:\/\/miro.medium.com\/v2\/resize:fill:96:96\/1*JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https:\/\/medium.com\/@kailashnagarajan?source=post_page---post_author_info--c08a27cc9012---------------------------------------)\n\n[![Kailash Nagarajan](https:\/\/miro.medium.com\/v2\/resize:fill:128:128\/1*JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https:\/\/medium.com\/@kailashnagarajan?source=post_page---post_author_info--c08a27cc9012---------------------------------------)\n\n[Written by Kailash Nagarajan](https:\/\/medium.com\/@kailashnagarajan?source=post_page---post_author_info--c08a27cc9012---------------------------------------)\n\n[22 followers](https:\/\/medium.com\/@kailashnagarajan\/followers?source=post_page---post_author_info--c08a27cc9012---------------------------------------)\n\n·[25 following](https:\/\/medium.com\/@kailashnagarajan\/following?source=post_page---post_author_info--c08a27cc9012---------------------------------------)\n\nThinker\/Tinkerer\/Maker\n## No responses yet\n[Help](https:\/\/help.medium.com\/hc\/en-us?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Status](https:\/\/status.medium.com\/?source=post_page-----c08a27cc9012---------------------------------------)\n\n[About](https:\/\/medium.com\/about?autoplay=1&source=post_page-----c08a27cc9012---------------------------------------)\n\n[Careers](https:\/\/medium.com\/jobs-at-medium\/work-at-medium-959d1a85284e?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Press](mailto:pressinquiries@medium.com)\n\n[Blog](https:\/\/blog.medium.com\/?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Privacy](https:\/\/policy.medium.com\/medium-privacy-policy-f03bf92035c9?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Rules](https:\/\/policy.medium.com\/medium-rules-30e5502c4eb4?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Terms](https:\/\/policy.medium.com\/medium-terms-of-service-9db0094a1e0f?source=post_page-----c08a27cc9012---------------------------------------)\n\n[Text to speech](https:\/\/speechify.com\/medium?source=post_page-----c08a27cc9012---------------------------------------)","attrs_readable_markdown":"[![Kailash Nagarajan](https:\/\/miro.medium.com\/v2\/resize:fill:64:64\/1*JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https:\/\/medium.com\/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------)\n\n3 min read Nov 10, 2018\n\n\\--\n\nOne of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into.\n\nBefore actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains?\n\nA function is anything that takes in an input and gives an output.Let’s consider a function f(x).\n\nPress enter or click to view image in full size\n\nSo for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this.\n\nA plot for f(x) = x²\n\nSimilarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc.\n\nNow, what is the definition of the domain of a given function? The **domain** is the set of all possible x-values which will make the **function** “work” and will output real f(x)-values.\n\nOne such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’.\n\n### Laplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’.\nHow do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain.\n\nThe formula for Laplace Transform\n\nF(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain.\n\nHere are the Laplace transforms of some basic functions.\n\nTable of Laplace Transforms of some basic functions.\n\n### Using Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain.\nOne of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations.\n\nFig1: Simple 1st order ODE\n\nThe above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below.\n\nPress enter or click to view image in full size\n\nLaplace transform of an nth order ODE\n\nThis amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations.\n\nIf you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function\n\nThank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from [here](https:\/\/www.youtube.com\/watch?v=OiNh2DswFt4&list=PL6D8DCFEAF1A468DD).","meta_canonical":null,"ml_categories_json":"","ml_types_json":"","ml_intent_types_json":"","meta_language":"en","attrs_author":"Kailash Nagarajan","attrs_publish_time":1541874550,"attrs_original_publish_time":1541874550,"attrs_is_republished":0,"attrs_nr_words":"657","attrs_boilerpipe_nr_words":"586","body_ext_links_number":3,"body_int_links_number":27,"meta_nofollow":0,"meta_noarchive":1,"props_was_rendered":0,"src_redirect":"","download_time_msec":462,"download_ttfb_msec":461,"download_size":28020}

3. Robots.txt Check

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Response:

4. Spam/Ban Check

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Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

1 month ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	1.1 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://medium.com/@kailashnagarajan/laplace-transforms-an-untold-story-c08a27cc9012
Last Crawled	2026-03-21 19:39:48 (1 month ago)
First Indexed	not set
HTTP Status Code	200
Content
Meta Title	Laplace Transforms: An Untold Story \| by Kailash Nagarajan \| Medium
Meta Description	Laplace Transforms: An Untold Story One of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way …
Meta Canonical	null
Boilerpipe Text	3 min read Nov 10, 2018 -- One of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into. Before actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains? A function is anything that takes in an input and gives an output.Let’s consider a function f(x). Press enter or click to view image in full size So for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this. A plot for f(x) = x² Similarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc. Now, what is the definition of the domain of a given function? The domain is the set of all possible x-values which will make the function “work” and will output real f(x)-values. One such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’. Laplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’. How do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain. The formula for Laplace Transform F(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain. Here are the Laplace transforms of some basic functions. Table of Laplace Transforms of some basic functions. Using Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain. One of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations. Fig1: Simple 1st order ODE The above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below. Press enter or click to view image in full size Laplace transform of an nth order ODE This amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations. If you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function Thank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from here .
Markdown	[Sitemap](https://medium.com/sitemap/sitemap.xml) [Open in app](https://play.google.com/store/apps/details?id=com.medium.reader&referrer=utm_source%3DmobileNavBar&source=post_page---top_nav_layout_nav-----------------------------------------) Sign up [Sign in](https://medium.com/m/signin?operation=login&redirect=https%3A%2F%2Fmedium.com%2F%40kailashnagarajan%2Flaplace-transforms-an-untold-story-c08a27cc9012&source=post_page---top_nav_layout_nav-----------------------global_nav------------------) [Medium Logo](https://medium.com/?source=post_page---top_nav_layout_nav-----------------------------------------) Get app [Write](https://medium.com/m/signin?operation=register&redirect=https%3A%2F%2Fmedium.com%2Fnew-story&source=---top_nav_layout_nav-----------------------new_post_topnav------------------) [Search](https://medium.com/search?source=post_page---top_nav_layout_nav-----------------------------------------) Sign up [Sign in](https://medium.com/m/signin?operation=login&redirect=https%3A%2F%2Fmedium.com%2F%40kailashnagarajan%2Flaplace-transforms-an-untold-story-c08a27cc9012&source=post_page---top_nav_layout_nav-----------------------global_nav------------------) ![](https://miro.medium.com/v2/resize:fill:64:64/1dmbNkD5D-u45r44go_cf0g.png) # Laplace Transforms: An Untold Story [![Kailash Nagarajan](https://miro.medium.com/v2/resize:fill:64:64/1JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https://medium.com/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------) [Kailash Nagarajan](https://medium.com/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------) 3 min read · Nov 10, 2018 \-- Listen Share One of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into. Before actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains? A function is anything that takes in an input and gives an output.Let’s consider a function f(x). Press enter or click to view image in full size ![]() So for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this. ![]() A plot for f(x) = x² Similarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc. Now, what is the definition of the domain of a given function? The domain is the set of all possible x-values which will make the function “work” and will output real f(x)-values. One such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’. ### Laplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’. How do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain. ![]() The formula for Laplace Transform F(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain. Here are the Laplace transforms of some basic functions. ![]() Table of Laplace Transforms of some basic functions. ### Using Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain. One of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations. ![]() Fig1: Simple 1st order ODE The above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below. Press enter or click to view image in full size ![]() Laplace transform of an nth order ODE This amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations. If you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function Thank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from [here](https://www.youtube.com/watch?v=OiNh2DswFt4&list=PL6D8DCFEAF1A468DD). 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Readable Markdown	[![Kailash Nagarajan](https://miro.medium.com/v2/resize:fill:64:64/1JwpUooNRGDGfMjx3Ctl5Dw.jpeg)](https://medium.com/@kailashnagarajan?source=post_page---byline--c08a27cc9012---------------------------------------) 3 min read Nov 10, 2018 \-- One of the major mathematical tool developed in the 18th century to make a lot of mathematical problems easier to solve and understand. Also, Changed the way Calculus was looked into. Before actually talking about Laplace transforms, There are a couple of questions to answer. What are functions? What is a domain? What is the need of changing domains? A function is anything that takes in an input and gives an output.Let’s consider a function f(x). Press enter or click to view image in full size So for every value of ‘x’, f(x) will have different values, For example, if x =2, f(x) will be 4 and if I plot a graph with f(x) on the y-axes of the plot and x on the x-axes of the plot. The plot will look something like this. A plot for f(x) = x² Similarly, you have so many different functions in mathematics, Trigonometric functions like sine, cosine, and tangent. Exponential Functions, Logarithmic functions, Hyperbolic Functions etc. Now, what is the definition of the domain of a given function? The domain* is the set of all possible x-values which will make the function “work” and will output real f(x)-values. One such domain is called time. We generally assume that all the values are changing wrt to the time domain. In the previous example f(x) was in the domain of ‘x’. Similarly, we can have a time domain ‘t’. ### Laplace Transforms. Why do we need Laplace transforms? To basically simplify the method of solving a lot of problems. To linearize the complex Differential and Integral equations into linear equations, By transforming from the time domain ‘t’ to the frequency domain,‘s’. How do I do that? It’s a very simple integral equation that takes us from the time domain to the frequency domain. The formula for Laplace Transform F(s) is the value of the function in the frequency domain and f(t) is the value of the function in the time domain. Here are the Laplace transforms of some basic functions. Table of Laplace Transforms of some basic functions. ### Using Inverse Laplace transform you can move back to the time domain from the frequency domain, So using Laplace and Inverse Laplace transforms you can move back and forth between time domain and frequency domain. One of the major usage of Laplace transforms is to convert complex differential equations to simple linear equations and help us solve them directly. Let us take an example and solve it so that you get a hang of Laplace transforms for differential equations. Fig1: Simple 1st order ODE The above example shows how Laplace transform can be used to solve a simple 1st order ordinary differential equation similarly, We can solve an ‘nth’ order differential equation using Laplace transform using the equation below. Press enter or click to view image in full size Laplace transform of an nth order ODE This amazing method that solves ODE’s easily helps us understand solution different dynamic systems also helps us explain the frequency response of different dynamic systems and also making the time response analysis easier by simplifying the method to solve differential equations. If you are using MATLAB, A very simple command “laplace(f)” can help you find the Laplace transform of any given function or equation. The command “ilaplace(f)” gives you the Inverse Laplace transform of a given frequency domain function Thank you for reading, If you are interested in learning more about Laplace Transforms, you can learn from [here](https://www.youtube.com/watch?v=OiNh2DswFt4&list=PL6D8DCFEAF1A468DD).
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