Problem 51745. Solve an ODE: equation A

Created by ChrisR

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to solve the following ordinary differential equation: </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"y&quot;-xy = 0\"/></w:customXmlPr><w:r><w:t>\\frac{d^2y}{dx^2} – x y = 0</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"y(x0) = y0\"/></w:customXmlPr><w:r><w:t>y(x_0) = y_0</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"y(x1) = y1\"/></w:customXmlPr><w:r><w:t>y(x_1) = y_1</w:t></w:r></w:customXml><w:r><w:t>. The function should return the values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"y\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t> at the specified values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"x\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r><w:r><w:t>One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 118.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.2917px; transform-origin: 407px 59.2917px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 7.91667px; transform-origin: 209.417px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to solve the following ordinary differential equation: </span></span></div><div style="block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-16px"><img src="data:image/png;base64,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" alt="y&quot;-xy = 0" style="width: 74px; height: 37.5px;" width="74" height="37.5"></span></div><div style="block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; "><span style="">with </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="y(x0) = y0" style="width: 64.5px; height: 20px;" width="64.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="y(x1) = y1" style="width: 64.5px; height: 20px;" width="64.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; "><span style="">. The function should return the values of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">y</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 7.91667px; transform-origin: 80.9px 7.91667px; unicode-bidi: normal; "><span style=""> at the specified values of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style="">.</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.35px 7.91667px; transform-origin: 58.35px 7.91667px; unicode-bidi: normal; "><span style="">One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.</span></span></div></div></div>

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Last Solution submitted on Jan 23, 2022

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David Hill on 14 May 2021

Chris,
Can you loosen the tolerance to 1e-13. Cannot seem to get all my answers within tolerance.

ChrisR on 14 May 2021

The tight tolerances are meant to encourage analytical solutions. A tag and the problem title provide clues. Nevertheless, I've loosened the tolerance on the tests that solution 5760543 didn't pass.

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Problem 51745. Solve an ODE: equation A

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Problem 51745. Solve an ODE: equation A

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