Show that if $u$ is a solution of the initial value problem $u^{\prime\prime} = -x^2u$, $u(0) = 1$, $u^{\prime}(0) = 0$, then it is a solution of the integral equation
$$u(x) = 1 – \int_0^x (x-s)(s^2)u(s) ds.$$
Consider the sequence $u_n(x)$ to be the iterated solutions to this integral equation. Assuming this sequence converges uniformly so that a solution exists, show that the solution is unique.
I literally have no idea how to do this problem – for the first bit I don't know if I need to integrate through the ODE. For the second bit, I think Picard's Theorem needs to be used but I don't know how.
Best Answer
If you are asking uniqueness of solutions you can use Gronwall's inequality. For the existence case you can form a sequence $u_n(x)$ defined by $$u_{n+1}(x) = 1 - \int_0^x (x-s)(s^2)u_n(s) ds$$ and then you proceed to show that $u_n$ converges uniformly a function $u$ which is the solution of the problem.