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1. Shard Calculation

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Calculated Shard: 152 (from laksa154)

2. Crawled Status Check

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curl -X POST \
  'http://laksa152.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 FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations","crawl_time":1757873167,"first_indexed_time":0,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Laplace_transform_applied_to_differential_equations s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Laplace transform applied to differential equations - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":null,"attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages)\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/wikipedia.png) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Laplace+transform+applied+to+differential+equations \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Laplace+transform+applied+to+differential+equations \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPersonal tools\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Laplace+transform+applied+to+differential+equations \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Laplace+transform+applied+to+differential+equations \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPages for logged out editors [learn more](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction)\n\n- [Contributions](https:\/\/en.wikipedia.org\/wiki\/Special:MyContributions \"A list of edits made from this IP address [y]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Special:MyTalk \"Discussion about edits from this IP address [n]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations)\n- [1 Approach](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations#Approach)\n- [2 An example](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations#An_example)\n- [3 Bibliography](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform_applied_to_differential_equations#Bibliography)\n\nToggle the table of contents\n\n# Laplace transform applied to differential equations\n4 languages\n\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Application_de_la_transform%C3%A9e_de_Laplace_aux_%C3%A9quations_diff%C3%A9rentielles \"Application de la transformée de Laplace aux équations différentielles – French\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/M%C3%A9todo_das_transformadas_de_Laplace_para_resolver_equa%C3%A7%C3%B5es_diferencais \"Método das transformadas de Laplace para resolver equações diferencais – Portuguese\")\n- [Shqip](https:\/\/sq.wikipedia.org\/wiki\/Transformimi_i_Laplasit_i_zbatuar_n%C3%AB_ekuacione_diferenciale \"Transformimi i Laplasit i zbatuar në ekuacione diferenciale – Albanian\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Laplacetransformen_av_differentialekvationer \"Laplacetransformen av differentialekvationer – Swedish\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q2858841#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- 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[e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Laplace_transform_applied_to_differential_equations \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChangesLinked\/Laplace_transform_applied_to_differential_equations \"Recent changes in pages linked from this page [k]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_Upload_Wizard \"Upload files [u]\")\n- [Permanent link](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&oldid=1289751115 \"Permanent link to this revision of this page\")\n- [Page information](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&action=info \"More information about this page\")\n- [Cite this page](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CiteThisPage&page=Laplace_transform_applied_to_differential_equations&id=1289751115&wpFormIdentifier=titleform \"Information on how to cite this page\")\n- [Get shortened URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLaplace_transform_applied_to_differential_equations)\n- [Download QR code](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLaplace_transform_applied_to_differential_equations)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Laplace_transform_applied_to_differential_equations&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q2858841 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nIn [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"), the [Laplace transform](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform \"Laplace transform\") is a powerful [integral transform](https:\/\/en.wikipedia.org\/wiki\/Integral_transform \"Integral transform\") used to switch a function from the [time domain](https:\/\/en.wikipedia.org\/wiki\/Time_domain \"Time domain\") to the [s-domain](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform#s-domain_equivalent_circuits_and_impedances \"Laplace transform\"). The Laplace transform can be used in some cases to solve [linear differential equations](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation \"Linear differential equation\") with given [initial conditions](https:\/\/en.wikipedia.org\/wiki\/Initial_value_problem \"Initial value problem\").\n\n## Approach\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=1 \"Edit section: Approach\")\\]\n\nFirst consider the following property of the Laplace transform:\n\nL\n\n{\n\nf\n\n′\n\n}\n\n\\=\n\ns\n\nL\n\n{\n\nf\n\n}\n\n−\n\nf\n\n(\n\n0\n\n)\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f'\\\\}=s{\\\\mathcal {L}}\\\\{f\\\\}-f(0)}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f'\\\\}=s{\\\\mathcal {L}}\\\\{f\\\\}-f(0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/38c4778e0226d35ab990383602960a029c80af87)\n\nL\n\n{\n\nf\n\n″\n\n}\n\n\\=\n\ns\n\n2\n\nL\n\n{\n\nf\n\n}\n\n−\n\ns\n\nf\n\n(\n\n0\n\n)\n\n−\n\nf\n\n′\n\n(\n\n0\n\n)\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f''\\\\}=s^{2}{\\\\mathcal {L}}\\\\{f\\\\}-sf(0)-f'(0)}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f''\\\\}=s^{2}{\\\\mathcal {L}}\\\\{f\\\\}-sf(0)-f'(0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/864209046b4627ad57e2e380695e107c8f922d17)\n\nOne can prove by [induction](https:\/\/en.wikipedia.org\/wiki\/Mathematical_induction \"Mathematical induction\") that\n\nL\n\n{\n\nf\n\n(\n\nn\n\n)\n\n}\n\n\\=\n\ns\n\nn\n\nL\n\n{\n\nf\n\n}\n\n−\n\n∑\n\ni\n\n\\=\n\n1\n\nn\n\ns\n\nn\n\n−\n\ni\n\nf\n\n(\n\ni\n\n−\n\n1\n\n)\n\n(\n\n0\n\n)\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f^{(n)}\\\\}=s^{n}{\\\\mathcal {L}}\\\\{f\\\\}-\\\\sum \\_{i=1}^{n}s^{n-i}f^{(i-1)}(0)}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f^{(n)}\\\\}=s^{n}{\\\\mathcal {L}}\\\\{f\\\\}-\\\\sum \\_{i=1}^{n}s^{n-i}f^{(i-1)}(0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0ec78bb43e6fc1df7618075c4080cf7998d3524a)\n\nNow we consider the following differential equation:\n\n∑\n\ni\n\n\\=\n\n0\n\nn\n\na\n\ni\n\nf\n\n(\n\ni\n\n)\n\n(\n\nt\n\n)\n\n\\=\n\nϕ\n\n(\n\nt\n\n)\n\n{\\\\displaystyle \\\\sum \\_{i=0}^{n}a\\_{i}f^{(i)}(t)=\\\\phi (t)}\n\n![{\\\\displaystyle \\\\sum \\_{i=0}^{n}a\\_{i}f^{(i)}(t)=\\\\phi (t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/31d6f6b1b629b123eeb8e6ed4883fa25ce68e337)\n\nwith given initial conditions\n\nf\n\n(\n\ni\n\n)\n\n(\n\n0\n\n)\n\n\\=\n\nc\n\ni\n\n{\\\\displaystyle f^{(i)}(0)=c\\_{i}}\n\n![{\\\\displaystyle f^{(i)}(0)=c\\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fb5e84e0a4868e792e7c81795b2d4fd69656aaa9)\n\nUsing the [linearity](https:\/\/en.wikipedia.org\/wiki\/Linearity \"Linearity\") of the Laplace transform it is equivalent to rewrite the equation as\n\n∑\n\ni\n\n\\=\n\n0\n\nn\n\na\n\ni\n\nL\n\n{\n\nf\n\n(\n\ni\n\n)\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n{\\\\displaystyle \\\\sum \\_{i=0}^{n}a\\_{i}{\\\\mathcal {L}}\\\\{f^{(i)}(t)\\\\}={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}\n\n![{\\\\displaystyle \\\\sum \\_{i=0}^{n}a\\_{i}{\\\\mathcal {L}}\\\\{f^{(i)}(t)\\\\}={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8021ebd9e4c0f56a8fa2024416e7a38cc862e6ea)\n\nobtaining\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n∑\n\ni\n\n\\=\n\n0\n\nn\n\na\n\ni\n\ns\n\ni\n\n−\n\n∑\n\ni\n\n\\=\n\n1\n\nn\n\n∑\n\nj\n\n\\=\n\n1\n\ni\n\na\n\ni\n\ns\n\ni\n\n−\n\nj\n\nf\n\n(\n\nj\n\n−\n\n1\n\n)\n\n(\n\n0\n\n)\n\n\\=\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}\\\\sum \\_{i=0}^{n}a\\_{i}s^{i}-\\\\sum \\_{i=1}^{n}\\\\sum \\_{j=1}^{i}a\\_{i}s^{i-j}f^{(j-1)}(0)={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}\\\\sum \\_{i=0}^{n}a\\_{i}s^{i}-\\\\sum \\_{i=1}^{n}\\\\sum \\_{j=1}^{i}a\\_{i}s^{i-j}f^{(j-1)}(0)={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3967dd75c59758605c672a1ada0a95207d560afe)\n\nSolving the equation for L { f ( t ) } {\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}} ![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a0b803a31a5862892069b233f032b8b671f066df) and substituting f ( i ) ( 0 ) {\\\\displaystyle f^{(i)}(0)} ![{\\\\displaystyle f^{(i)}(0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/850bbeca83f6c23f3a32e9d5d5d37b2ed6ba7246) with c i {\\\\displaystyle c\\_{i}} ![{\\\\displaystyle c\\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/01acb7953ba52c2aa44264b5d0f8fd223aa178a2) one obtains\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n\\+\n\n∑\n\ni\n\n\\=\n\n1\n\nn\n\n∑\n\nj\n\n\\=\n\n1\n\ni\n\na\n\ni\n\ns\n\ni\n\n−\n\nj\n\nc\n\nj\n\n−\n\n1\n\n∑\n\ni\n\n\\=\n\n0\n\nn\n\na\n\ni\n\ns\n\ni\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\frac {{\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}+\\\\sum \\_{i=1}^{n}\\\\sum \\_{j=1}^{i}a\\_{i}s^{i-j}c\\_{j-1}}{\\\\sum \\_{i=0}^{n}a\\_{i}s^{i}}}}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\frac {{\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}+\\\\sum \\_{i=1}^{n}\\\\sum \\_{j=1}^{i}a\\_{i}s^{i-j}c\\_{j-1}}{\\\\sum \\_{i=0}^{n}a\\_{i}s^{i}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9afee08e8dd3015de43b631c95bd3874f7b8999f)\n\nThe solution for *f*(*t*) is obtained by applying the [inverse Laplace transform](https:\/\/en.wikipedia.org\/wiki\/Inverse_Laplace_transform \"Inverse Laplace transform\") to L { f ( t ) } . {\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}.} ![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cadc4cf50d84eb251df1f69f7038219b2a80d17f)\n\nNote that if the initial conditions are all zero, i.e.\n\nf\n\n(\n\ni\n\n)\n\n(\n\n0\n\n)\n\n\\=\n\nc\n\ni\n\n\\=\n\n0\n\n∀\n\ni\n\n∈\n\n{\n\n0\n\n,\n\n1\n\n,\n\n2\n\n,\n\n.\n\n.\n\n.\n\nn\n\n}\n\n{\\\\displaystyle f^{(i)}(0)=c\\_{i}=0\\\\quad \\\\forall i\\\\in \\\\{0,1,2,...\\\\ n\\\\}}\n\n![{\\\\displaystyle f^{(i)}(0)=c\\_{i}=0\\\\quad \\\\forall i\\\\in \\\\{0,1,2,...\\\\ n\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5d9deced0bee2f122793751b0e970aa54dc03ff2)\n\nthen the formula simplifies to\n\nf\n\n(\n\nt\n\n)\n\n\\=\n\nL\n\n−\n\n1\n\n{\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n∑\n\ni\n\n\\=\n\n0\n\nn\n\na\n\ni\n\ns\n\ni\n\n}\n\n{\\\\displaystyle f(t)={\\\\mathcal {L}}^{-1}\\\\left\\\\{{{\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\} \\\\over \\\\sum \\_{i=0}^{n}a\\_{i}s^{i}}\\\\right\\\\}}\n\n![{\\\\displaystyle f(t)={\\\\mathcal {L}}^{-1}\\\\left\\\\{{{\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\} \\\\over \\\\sum \\_{i=0}^{n}a\\_{i}s^{i}}\\\\right\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9ab37a095ec0f07502a1b8d43b7679b59d346a78)\n\n## An example\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=2 \"Edit section: An example\")\\]\n\nWe want to solve\n\nf\n\n″\n\n(\n\nt\n\n)\n\n\\+\n\n4\n\nf\n\n(\n\nt\n\n)\n\n\\=\n\nsin\n\n⁡\n\n(\n\n2\n\nt\n\n)\n\n{\\\\displaystyle f''(t)+4f(t)=\\\\sin(2t)}\n\n![{\\\\displaystyle f''(t)+4f(t)=\\\\sin(2t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7108de2c6ab151cd8129e2c610069e6419e23953)\n\nwith initial conditions *f*(0) = 0 and *f′*(0)=0.\n\nWe note that\n\nϕ\n\n(\n\nt\n\n)\n\n\\=\n\nsin\n\n⁡\n\n(\n\n2\n\nt\n\n)\n\n{\\\\displaystyle \\\\phi (t)=\\\\sin(2t)}\n\n![{\\\\displaystyle \\\\phi (t)=\\\\sin(2t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5aa8b3a81124fbd190c768f1909fac24087e8f55)\n\nand we get\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\n2\n\ns\n\n2\n\n\\+\n\n4\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}={\\\\frac {2}{s^{2}+4}}}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}={\\\\frac {2}{s^{2}+4}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6a281ca6ea972cb29aa64da883a645a06ca33c6f)\n\nThe equation is then equivalent to\n\ns\n\n2\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n−\n\ns\n\nf\n\n(\n\n0\n\n)\n\n−\n\nf\n\n′\n\n(\n\n0\n\n)\n\n\\+\n\n4\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\nL\n\n{\n\nϕ\n\n(\n\nt\n\n)\n\n}\n\n{\\\\displaystyle s^{2}{\\\\mathcal {L}}\\\\{f(t)\\\\}-sf(0)-f'(0)+4{\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}\n\n![{\\\\displaystyle s^{2}{\\\\mathcal {L}}\\\\{f(t)\\\\}-sf(0)-f'(0)+4{\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\mathcal {L}}\\\\{\\\\phi (t)\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0745e1fe37e32503e9d1146f5e76e92bd0262861)\n\nWe deduce\n\nL\n\n{\n\nf\n\n(\n\nt\n\n)\n\n}\n\n\\=\n\n2\n\n(\n\ns\n\n2\n\n\\+\n\n4\n\n)\n\n2\n\n{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\frac {2}{(s^{2}+4)^{2}}}}\n\n![{\\\\displaystyle {\\\\mathcal {L}}\\\\{f(t)\\\\}={\\\\frac {2}{(s^{2}+4)^{2}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a2d01da41841d414a0aa084e23fe9b12bc74a358)\n\nNow we apply the Laplace inverse transform to get\n\nf\n\n(\n\nt\n\n)\n\n\\=\n\n1\n\n8\n\nsin\n\n⁡\n\n(\n\n2\n\nt\n\n)\n\n−\n\nt\n\n4\n\ncos\n\n⁡\n\n(\n\n2\n\nt\n\n)\n\n{\\\\displaystyle f(t)={\\\\frac {1}{8}}\\\\sin(2t)-{\\\\frac {t}{4}}\\\\cos(2t)}\n\n![{\\\\displaystyle f(t)={\\\\frac {1}{8}}\\\\sin(2t)-{\\\\frac {t}{4}}\\\\cos(2t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87fc0df5945678fb828bb87086d5cb89efdd27ad)\n\n## Bibliography\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=3 \"Edit section: Bibliography\")\\]\n\n- A. D. Polyanin, *Handbook of Linear Partial Differential Equations for Engineers and Scientists*, Chapman & Hall\/CRC Press, Boca Raton, 2002. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [1-58488-299-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/1-58488-299-9 \"Special:BookSources\/1-58488-299-9\")\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Laplace_transform_applied_to_differential_equations&oldid=1289751115>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Integral transforms](https:\/\/en.wikipedia.org\/wiki\/Category:Integral_transforms \"Category:Integral transforms\")\n- [Differential equations](https:\/\/en.wikipedia.org\/wiki\/Category:Differential_equations \"Category:Differential equations\")\n- [Differential calculus](https:\/\/en.wikipedia.org\/wiki\/Category:Differential_calculus \"Category:Differential calculus\")\n- [Ordinary differential equations](https:\/\/en.wikipedia.org\/wiki\/Category:Ordinary_differential_equations \"Category:Ordinary differential equations\")\n- [Laplace transforms](https:\/\/en.wikipedia.org\/wiki\/Category:Laplace_transforms \"Category:Laplace transforms\")\n\n- This page was last edited on 10 May 2025, at 16:23 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. 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The Laplace transform can be used in some cases to solve [linear differential equations](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") with given [initial conditions](https://en.wikipedia.org/wiki/Initial_value_problem "Initial value problem"). ## Approach \[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=1 "Edit section: Approach")\] First consider the following property of the Laplace transform: L { f ′ } \= s L { f } − f ( 0 ) {\\displaystyle {\\mathcal {L}}\\{f'\\}=s{\\mathcal {L}}\\{f\\}-f(0)} ![{\\displaystyle {\\mathcal {L}}\\{f'\\}=s{\\mathcal {L}}\\{f\\}-f(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38c4778e0226d35ab990383602960a029c80af87) L { f ″ } \= s 2 L { f } − s f ( 0 ) − f ′ ( 0 ) {\\displaystyle {\\mathcal {L}}\\{f''\\}=s^{2}{\\mathcal {L}}\\{f\\}-sf(0)-f'(0)} ![{\\displaystyle {\\mathcal {L}}\\{f''\\}=s^{2}{\\mathcal {L}}\\{f\\}-sf(0)-f'(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864209046b4627ad57e2e380695e107c8f922d17) One can prove by [induction](https://en.wikipedia.org/wiki/Mathematical_induction "Mathematical induction") that L { f ( n ) } \= s n L { f } − ∑ i \= 1 n s n − i f ( i − 1 ) ( 0 ) {\\displaystyle {\\mathcal {L}}\\{f^{(n)}\\}=s^{n}{\\mathcal {L}}\\{f\\}-\\sum \_{i=1}^{n}s^{n-i}f^{(i-1)}(0)} ![{\\displaystyle {\\mathcal {L}}\\{f^{(n)}\\}=s^{n}{\\mathcal {L}}\\{f\\}-\\sum \_{i=1}^{n}s^{n-i}f^{(i-1)}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec78bb43e6fc1df7618075c4080cf7998d3524a) Now we consider the following differential equation: ∑ i \= 0 n a i f ( i ) ( t ) \= ϕ ( t ) {\\displaystyle \\sum \_{i=0}^{n}a\_{i}f^{(i)}(t)=\\phi (t)} ![{\\displaystyle \\sum \_{i=0}^{n}a\_{i}f^{(i)}(t)=\\phi (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d6f6b1b629b123eeb8e6ed4883fa25ce68e337) with given initial conditions f ( i ) ( 0 ) \= c i {\\displaystyle f^{(i)}(0)=c\_{i}} ![{\\displaystyle f^{(i)}(0)=c\_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5e84e0a4868e792e7c81795b2d4fd69656aaa9) Using the [linearity](https://en.wikipedia.org/wiki/Linearity "Linearity") of the Laplace transform it is equivalent to rewrite the equation as ∑ i \= 0 n a i L { f ( i ) ( t ) } \= L { ϕ ( t ) } {\\displaystyle \\sum \_{i=0}^{n}a\_{i}{\\mathcal {L}}\\{f^{(i)}(t)\\}={\\mathcal {L}}\\{\\phi (t)\\}} ![{\\displaystyle \\sum \_{i=0}^{n}a\_{i}{\\mathcal {L}}\\{f^{(i)}(t)\\}={\\mathcal {L}}\\{\\phi (t)\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8021ebd9e4c0f56a8fa2024416e7a38cc862e6ea) obtaining L { f ( t ) } ∑ i \= 0 n a i s i − ∑ i \= 1 n ∑ j \= 1 i a i s i − j f ( j − 1 ) ( 0 ) \= L { ϕ ( t ) } {\\displaystyle {\\mathcal {L}}\\{f(t)\\}\\sum \_{i=0}^{n}a\_{i}s^{i}-\\sum \_{i=1}^{n}\\sum \_{j=1}^{i}a\_{i}s^{i-j}f^{(j-1)}(0)={\\mathcal {L}}\\{\\phi (t)\\}} ![{\\displaystyle {\\mathcal {L}}\\{f(t)\\}\\sum \_{i=0}^{n}a\_{i}s^{i}-\\sum \_{i=1}^{n}\\sum \_{j=1}^{i}a\_{i}s^{i-j}f^{(j-1)}(0)={\\mathcal {L}}\\{\\phi (t)\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3967dd75c59758605c672a1ada0a95207d560afe) Solving the equation for L { f ( t ) } {\\displaystyle {\\mathcal {L}}\\{f(t)\\}} ![{\\displaystyle {\\mathcal {L}}\\{f(t)\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b803a31a5862892069b233f032b8b671f066df) and substituting f ( i ) ( 0 ) {\\displaystyle f^{(i)}(0)} ![{\\displaystyle f^{(i)}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850bbeca83f6c23f3a32e9d5d5d37b2ed6ba7246) with c i {\\displaystyle c\_{i}} ![{\\displaystyle c\_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2) one obtains L { f ( t ) } \= L { ϕ ( t ) } \+ ∑ i \= 1 n ∑ j \= 1 i a i s i − j c j − 1 ∑ i \= 0 n a i s i {\\displaystyle {\\mathcal {L}}\\{f(t)\\}={\\frac {{\\mathcal {L}}\\{\\phi (t)\\}+\\sum \_{i=1}^{n}\\sum \_{j=1}^{i}a\_{i}s^{i-j}c\_{j-1}}{\\sum \_{i=0}^{n}a\_{i}s^{i}}}} ![{\\displaystyle {\\mathcal {L}}\\{f(t)\\}={\\frac {{\\mathcal {L}}\\{\\phi (t)\\}+\\sum \_{i=1}^{n}\\sum \_{j=1}^{i}a\_{i}s^{i-j}c\_{j-1}}{\\sum \_{i=0}^{n}a\_{i}s^{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afee08e8dd3015de43b631c95bd3874f7b8999f) The solution for f(t) is obtained by applying the [inverse Laplace transform](https://en.wikipedia.org/wiki/Inverse_Laplace_transform "Inverse Laplace transform") to L { f ( t ) } . {\\displaystyle {\\mathcal {L}}\\{f(t)\\}.} ![{\\displaystyle {\\mathcal {L}}\\{f(t)\\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cadc4cf50d84eb251df1f69f7038219b2a80d17f) Note that if the initial conditions are all zero, i.e. f ( i ) ( 0 ) \= c i \= 0 ∀ i ∈ { 0 , 1 , 2 , . . . n } {\\displaystyle f^{(i)}(0)=c\_{i}=0\\quad \\forall i\\in \\{0,1,2,...\\ n\\}} ![{\\displaystyle f^{(i)}(0)=c\_{i}=0\\quad \\forall i\\in \\{0,1,2,...\\ n\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9deced0bee2f122793751b0e970aa54dc03ff2) then the formula simplifies to f ( t ) \= L − 1 { L { ϕ ( t ) } ∑ i \= 0 n a i s i } {\\displaystyle f(t)={\\mathcal {L}}^{-1}\\left\\{{{\\mathcal {L}}\\{\\phi (t)\\} \\over \\sum \_{i=0}^{n}a\_{i}s^{i}}\\right\\}} ![{\\displaystyle f(t)={\\mathcal {L}}^{-1}\\left\\{{{\\mathcal {L}}\\{\\phi (t)\\} \\over \\sum \_{i=0}^{n}a\_{i}s^{i}}\\right\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab37a095ec0f07502a1b8d43b7679b59d346a78) ## An example \[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=2 "Edit section: An example")\] We want to solve f ″ ( t ) \+ 4 f ( t ) \= sin ⁡ ( 2 t ) {\\displaystyle f''(t)+4f(t)=\\sin(2t)} ![{\\displaystyle f''(t)+4f(t)=\\sin(2t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7108de2c6ab151cd8129e2c610069e6419e23953) with initial conditions f(0) = 0 and f′(0)=0. We note that ϕ ( t ) \= sin ⁡ ( 2 t ) {\\displaystyle \\phi (t)=\\sin(2t)} ![{\\displaystyle \\phi (t)=\\sin(2t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa8b3a81124fbd190c768f1909fac24087e8f55) and we get L { ϕ ( t ) } \= 2 s 2 \+ 4 {\\displaystyle {\\mathcal {L}}\\{\\phi (t)\\}={\\frac {2}{s^{2}+4}}} ![{\\displaystyle {\\mathcal {L}}\\{\\phi (t)\\}={\\frac {2}{s^{2}+4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a281ca6ea972cb29aa64da883a645a06ca33c6f) The equation is then equivalent to s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) \+ 4 L { f ( t ) } \= L { ϕ ( t ) } {\\displaystyle s^{2}{\\mathcal {L}}\\{f(t)\\}-sf(0)-f'(0)+4{\\mathcal {L}}\\{f(t)\\}={\\mathcal {L}}\\{\\phi (t)\\}} ![{\\displaystyle s^{2}{\\mathcal {L}}\\{f(t)\\}-sf(0)-f'(0)+4{\\mathcal {L}}\\{f(t)\\}={\\mathcal {L}}\\{\\phi (t)\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0745e1fe37e32503e9d1146f5e76e92bd0262861) We deduce L { f ( t ) } \= 2 ( s 2 \+ 4 ) 2 {\\displaystyle {\\mathcal {L}}\\{f(t)\\}={\\frac {2}{(s^{2}+4)^{2}}}} ![{\\displaystyle {\\mathcal {L}}\\{f(t)\\}={\\frac {2}{(s^{2}+4)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d01da41841d414a0aa084e23fe9b12bc74a358) Now we apply the Laplace inverse transform to get f ( t ) \= 1 8 sin ⁡ ( 2 t ) − t 4 cos ⁡ ( 2 t ) {\\displaystyle f(t)={\\frac {1}{8}}\\sin(2t)-{\\frac {t}{4}}\\cos(2t)} ![{\\displaystyle f(t)={\\frac {1}{8}}\\sin(2t)-{\\frac {t}{4}}\\cos(2t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87fc0df5945678fb828bb87086d5cb89efdd27ad) ## Bibliography \[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform_applied_to_differential_equations&action=edit&section=3 "Edit section: Bibliography")\] - A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [1-58488-299-9](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-299-9 "Special:BookSources/1-58488-299-9") ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Laplace_transform_applied_to_differential_equations&oldid=1289751115>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Integral transforms](https://en.wikipedia.org/wiki/Category:Integral_transforms "Category:Integral transforms") - [Differential equations](https://en.wikipedia.org/wiki/Category:Differential_equations "Category:Differential equations") - [Differential calculus](https://en.wikipedia.org/wiki/Category:Differential_calculus "Category:Differential calculus") - [Ordinary differential equations](https://en.wikipedia.org/wiki/Category:Ordinary_differential_equations "Category:Ordinary differential equations") - [Laplace transforms](https://en.wikipedia.org/wiki/Category:Laplace_transforms "Category:Laplace transforms") - This page was last edited on 10 May 2025, at 16:23 (UTC). - Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. 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