Problem 51783. Solve an ODE: equation B

Created by ChrisR

Solve Later

Write a function to solve the following ordinary differential equation:  
with  and either  or . Along with y1, one of y0 and yp0 will be assigned numerical values, and the other will be NaN. The function should return the values of y at the specified values of x.
One of the applications for this equation is in groundwater. For  and  the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable x is the normalized distance from the well, and y 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.  

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Last Solution submitted on May 09, 2025

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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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