Problem 51783. Solve an ODE: equation B

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>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; + (1/x) y' - (a^2 + p^2/x^2) y = 0\"/></w:customXmlPr><w:r><w:t>\\frac{d^2y}{dx^2} +\\frac{1}{x} \\frac{dy}{dx}-\\left(a^2+\\frac{p^2}{x^2}\\right)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(x1) = y1\"/></w:customXmlPr><w:r><w:t>y(x_1) = y_1</w:t></w:r></w:customXml><w:r><w:t> and either </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> or </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) = y'0\"/></w:customXmlPr><w:r><w:t>y\\prime(x_0) = y\\prime_0</w:t></w:r></w:customXml><w:r><w:t>. Along with </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y1</w:t></w:r><w:r><w:t>, one of </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>y0</w:t></w:r><w:r><w:t> and </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>yp0</w:t></w:r><w:r><w:t> will be assigned numerical values, and the other will be </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>NaN</w:t></w:r><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: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: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>One of the applications for this equation is in groundwater. For </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=\"a = 1\"/></w:customXmlPr><w:r><w:t>a = 1</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=\"p = 0\"/></w:customXmlPr><w:r><w:t>p = 0</w:t></w:r></w:customXml><w:r><w:t> the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable </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> is the normalized distance from the well, 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\"/></w:customXmlPr><w:r><w:t>y</w:t></w:r></w:customXml><w:r><w:t> is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.</w:t></w:r><w:r><w:t> </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: 213.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 106.625px; transform-origin: 407px 106.625px; 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: 211.358px 7.91667px; transform-origin: 211.358px 7.91667px; unicode-bidi: normal; ">Write a function to solve the following ordinary differential equation: </div><div style="block-size: 39px; 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 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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alt="y&quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0" style="width: 181.5px; height: 39px;" width="181.5" height="39"></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; ">with <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 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: 35.0083px 7.91667px; transform-origin: 35.0083px 7.91667px; unicode-bidi: normal; "> and either <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 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: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; "> or <img src="data:image/png;base64,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" alt="y'(x0) = y'0" style="width: 74px; height: 20px;" width="74" height="20"><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: 38.1167px 7.91667px; transform-origin: 38.1167px 7.91667px; unicode-bidi: normal; ">. Along with <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: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; ">y1<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: 25.275px 7.91667px; transform-origin: 25.275px 7.91667px; unicode-bidi: normal; ">, one of <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: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; ">y0<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; "> and <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: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; ">yp0<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: 109.708px 7.91667px; transform-origin: 109.708px 7.91667px; unicode-bidi: normal; "> will be assigned numerical values, and the other will be <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: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; ">NaN<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; ">. The function should return the values of y<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; "> at the specified values of x<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; ">.</div><div style="block-size: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 194.483px 7.91667px; transform-origin: 194.483px 7.91667px; unicode-bidi: normal; ">One of the applications for this equation is in groundwater. For <img src="data:image/png;base64,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" alt="a = 1" style="width: 36.5px; height: 18px;" width="36.5" height="18"><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; "> and <img src="data:image/png;base64,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" alt="p = 0" style="width: 38px; height: 18px;" width="38" height="18"><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: 136.567px 7.91667px; transform-origin: 136.567px 7.91667px; unicode-bidi: normal; "> the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable x<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: 141.975px 7.91667px; transform-origin: 141.975px 7.91667px; unicode-bidi: normal; "> is the normalized distance from the well, and y<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: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; "> is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.<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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; "> </div></div></div>

Solve

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Last Solution submitted on Nov 11, 2021

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2 Comments

David Hill on 17 May 2021

Chris,
I solved the first test suite symbolically to make sure I was on the right path an got:
[1.0, 6.4839432567518947511762312927844, 9.0049224364695494006414430797622, 8.992494072596270056922480305016, 6.9752731606055591524878412503205, 3.6901871946858786387135688683041, 0].
I am not sure why it would be wrong.
syms y(x)
Dy = diff(y);
a=1;p=0;
X=linspace(1,4,7);
ode=diff(y,x,2)==-Dy/x+(p^2/x^2-a^2)*y;
conds=[y(1)==1, y(4)==0];
ySol(x)=simplify(dsolve(ode,conds));
Y=vpa(ySol(X));

ChrisR on 17 May 2021

Sorry David. I made a mistake with the signs in the last term of the equation. I "modified" the problem description (quotes intentional). This version should work, but please let me know if it doesn't.

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Problem 51783. Solve an ODE: equation B

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Problem 51783. Solve an ODE: equation B

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