Picard's Theorem and Second order ODEs
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.
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$\begingroup$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.
$\endgroup$ 4 $\begingroup$For the first bit, you want to use the same method used to derive Picard's iterative approximation method, but twice. And then you want and then use this trick to turn your double integral into a single integral:
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