How to solve the boundary value problem for $\frac{d^2 y}{dx^2}+xy=1 $

Question

How to solve the boundary value problem given below:

$ \frac{d^2 y}{dx^2}+xy=1 $ , with the conditions $ y(2)=5 $ and $y(9)=6 $ .

I know the problem is given Dirichlet Boundary condition but I could not find a suitable method to get the solution.

Is there a way to solve it?

MrYouMath · Answer 1 · 2017-09-24T12:20:58.897

1

This inhomogeneous ODE is related to the Airy Differential equation. It cannot be solved, as far as I know, by analytical methods using elementary functions.

Transform the ODE into the first-order form and try to solve it using the shooting method.

Another method would be to use perturbation theory to derive an approximate expression for the solution.

EDIT: As Dr. Sonnhard Graubner implied the solution looks terrible, which you can decide by your self by using Wolfram Alpha.

edited Sep 24 '17 at 12:20

answered Sep 24 '17 at 12:07

MrYouMath

15,833

If I try using Laplace's transform , would it be applicable ? If not then how to get solution ? – MAS Sep 24 '17 at 12:08
Laplace Transform methods will also not help you solve this ODE. You can only solve it using numerical methods or by expressing the solution in terms of the $Ai(x)$ and $Bi(x)$ functions (see the link that I provided). – MrYouMath Sep 24 '17 at 12:10
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the symbolic solution by Mathematica Looks terrible – Dr. Sonnhard Graubner Sep 24 '17 at 12:11
This will be easy if we use Linear shooting method using 2 initial value problem . But I am unable to compose it – MAS Sep 24 '17 at 12:26
Even using Airy functions, this is a big, big, big monster – Claude Leibovici Sep 24 '17 at 13:31

How to solve the boundary value problem for $\frac{d^2 y}{dx^2}+xy=1 $

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