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?

Answer 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.