Problem 54000. Solve an ODE: first-order linear equation

Created by ChrisR

Solve Later

{"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>In the fall semester of [redacted], my professor for a course on ordinary differential equations said, “There are two groups of people in the world: those who can solve first-order linear ordinary differential equations and those who can’t. By the end of this lecture, you will be in the first group.” </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>I clearly remember appreciating the cleverness of the integrating factor that we learned that day. I also recall the professor exhorting us not to memorize the final formula but to remember the approach.</w:t></w:r></w:p><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 an equation of the form</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' + p(x) y = q(x)\"/></w:customXmlPr><w:r><w:t>y\\prime+p(x)y = q(x)</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 the condition </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>. 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:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Although MATLAB ODE functions are allowed, I would like someone to demonstrate how to implement the analytical approach using an integrating factor in MATLAB. Those who can will received the coveted unscored bonus points. </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: 256px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 128px; transform-origin: 407px 128px; vertical-align: baseline; "><div style="block-size: 63px; 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 31.5px; text-align: left; transform-origin: 384px 31.5px; 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: 383.767px 8px; transform-origin: 383.767px 8px; unicode-bidi: normal; "><span style="">In the fall semester of [redacted], my professor for a course on ordinary differential equations said, “There are two groups of people in the world: those who can solve first-order linear ordinary differential equations and those who can’t. By the end of this lecture, you will be in the first group.” </span></span></div><div style="block-size: 42px; 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 21px; text-align: left; transform-origin: 384px 21px; 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: 378.725px 8px; transform-origin: 378.725px 8px; unicode-bidi: normal; "><span style="">I clearly remember appreciating the cleverness of the integrating factor that we learned that day. I also recall the professor exhorting us not to memorize the final formula but to remember the approach.</span></span></div><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: 148.45px 8px; transform-origin: 148.45px 8px; unicode-bidi: normal; "><span style="">Write a function to solve an equation of the form</span></span></div><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="vertical-align:-5px"><img src="data:image/png;base64,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" alt="y' + p(x) y = q(x)" style="width: 113px; height: 18.5px;" width="113" height="18.5"></span></div><div style="block-size: 22px; 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 11px; text-align: left; transform-origin: 384px 11px; 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: 56.0167px 8px; transform-origin: 56.0167px 8px; unicode-bidi: normal; "><span style="">with the condition </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: 128.742px 8px; transform-origin: 128.742px 8px; 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 8px; transform-origin: 80.9px 8px; 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 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 42px; 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 21px; text-align: left; transform-origin: 384px 21px; 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: 361.233px 8px; transform-origin: 361.233px 8px; unicode-bidi: normal; "><span style="">Although MATLAB ODE functions are allowed, I would like someone to demonstrate how to implement the analytical approach using an integrating factor in MATLAB. Those who can will received the coveted unscored bonus points. </span></span></div></div></div>

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Last Solution submitted on Nov 20, 2022

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Solution 7507275

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ChrisR on 10 Feb 2022

Thanks Tim. I didn't know about that option for that function. This solution has the elegance that I was seeking.

Tim on 10 Feb 2022

I didn't know about it either--it was suggested by an error message that appeared when the option was absent, but I don't really understand why it needs to be there.

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Problem 54000. Solve an ODE: first-order linear equation

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Problem 54000. Solve an ODE: first-order linear equation

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