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,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAoCAYAAADZs5l2AAAGMklEQVR4Xu2bOastVRCF7/sDzpEYOQSCooEDiAYaOCaKgooGDxQnxEDEETFwFk3EEQxeoDhgYOAcKKgIooGiYODAjYwc8Qfo+qAL6tTd3b337nu6+57bBxbnvdO9p6q1q1bt7ntga/nsewsc2PcWWAywtZBgIcFCgoUDW6OT4FEZ/S3hu5GMf7fG+VL4YqTx9uQwY6aDF2Shz4Q3RrTU0RrreeGdkccdcYnDhxqLBBBgW3hq+JSLe4AIHwoPCe8Xt94HDcYgwc2y443CmRPa83iN/Y1whvDrhPOY5dDrJgHG/0U4bwZ5GX1wgXDxLD0x4aTWTQLSAESYg+FJC78L1y76YJVx6ySBRYHLZpSLISVpacrUNOGeTw+9ThJQDt4vrHOMUoNeqgbvCacLY5WppXMc/f51OuhnrQbMIRWYYS0l3KMfpqhURndwzoA5JMBwJws/Cn+ETgn5xyauWSooNfa56uvfxC5tGydnjfGeOZIzZx3Y5jchVjfmn9S1nH5bQzUDXiLcKhzZ9HSLvl8Ovf7ZXP9I39c5ktD+cyHVJk6MEvIK4aLmwl/69qWcpRUuPyY8kLWy9ps4MzhLOKqiH9Y19JMieapPiH+VcJNwQnPDm/q+JtzM4dvVQrRb9jz7IgEs+6qZBKXe2YKPBggtiMLnGHeNcuxJIac0PE33kZ9Z3OtNX0YeCHCh8HUzTg6p+hZvRutbe6qf//o6z7iecmRfM5sz90U94+1WJcJzDOEHSTk1pbhLSOANYEZ+UT++KzwrnNhnocLrNrcacUgUGfr5RB2U6hFLr4ydSrHmI78Rs+eZQwITU3SaCsfs1n/CwmpJgJFJC0QdwvVJQtQh2YtrubF2bkPHHdqeaEia5OQzlrhUPXcIVSI8hwRM3sIRE2Ag7xiE1pWCL7lqDW3tGDMn9Jsoys2z9Fs7t6FOHNoe7fRS0wkawQtEojH2j5qN203HtD5JzSWBn4APo4Shy4UoVmoNbYKSyXflN5x/p4AgOijcLhwh3BaMkzK86ZjctQ913m61Rzt923TmN4g9F4lCF589LlwvnCIgMLHVDjLkGsJPwB+7pqIA87RDmVKh4sfpKi8tMvkcSCqBHDFSRSdYysldu28/ZnWQIo9VY2gmCM8nFQVs03r7szHvFc4XVg7KSgwRJ8BAdBijABMzZ5acE/hKhD5SuY/fLVpElW3iqC+NkFtJZzX5c6rqwAgR0/JxuvCp4KMAdvypaeB/N3G5ozopIUGcAAN1PZqFNB+3kCTFcvqnzbZAecknpXYtnEeCGfFSusWPhyNrzxumqg5s/j4tY5vXhEOCf1HHNgNnN5HotpFXNEUJCWwCHErgXBR818ENzqLGzynx/DsHKV3Ab+cIlFamklOpxnZqW6lUm6ZSpJ3iN58uSQmph2F2uJYiuqXCFduVkMA7BwL0ObfrYQ2LoRT8QThMgDC+HPTnBSheQp5dt2tdJGjTIhgIMdk39ykcnDumT0mxSqAPc3QXCVaulZCAAbockFoEwpGoYSLG7rGJ2v/jwY0/IeMef30ICQiHTzQRJdfoc7uvy8mjkcAr0z4DEQ1eDbucNna48bf+TRkTH+sSKe5rOo/Xa0lArnxEiEfffWuY23VIQCRri2ZrjQSkg0Mdg7cZywRfjAa1xiW6EAa70kE83jbFTM28l182tXV0CXITzl3pYEVU56aDIS9q7vbbvpYqIgn88XZcF236hGwtKcdshyh+Ruh6bd8EfIoEJqpXNkkOCXbDiZCI+pS3joe+0dO2SBOusTTiftLLbkWiMZ3ux2KHU5n1PUr3pXJ8xmCpdMXvKRKQOw8KPO16u3HeK/pOnUuXGAQyUdcSioYQwUIiO9ufDlpp5CMEazk1w3Al6xjjXhzJWQnPB54W7hI4+EkdzKXmY7rAVw9Wre2IECkSROW+22/n4sShTwZtQXY6iNEoIxGtfqcQgfbi3xn4B2k4uUSMc7+lb9LGgw1L7KBrx0u2KRKYMqecgoVzNSILZYdsC4cLHwib8jeHbJSHm93/XOW66OOGxjbw4HshqSVyNEEq3Cy/bZAFFhJskDNrl7KQoNZyG9RuIcEGObN2KQsJai23Qe0WEmyQM2uXspCg1nIb1O5/ComHOGWac4gAAAAASUVORK5CYII=" 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>

Solve

Solution Stats

16 Solutions
3 Solvers

Last Solution submitted on Nov 20, 2022

Last 200 Solutions

Problem Comments

Solution Comments

Solution 7507275

2 Comments

2 Comments

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.

Problem Recent Solvers3

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

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

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers3

Suggested Problems

More from this Author192

Problem Tags

Community Treasure Hunt

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

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers3

Suggested Problems

More from this Author192

Problem Tags

Community Treasure Hunt

Players