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**sakodo** Hi guys I am currently stuck at a question.

i) Show that if $\displaystyle x(t)$ satisfies the integral equation

$\displaystyle x(t)=a+bt+ \int_0^t (t-s)f(x(s)) ds$

then $\displaystyle x(t)$ is a solution to the initial value problem

$\displaystyle x''(t)=f(x(t))$ for $\displaystyle t>0$, with $\displaystyle x(0)=a , \ x'(0)=b.$

ii) Prove the converse of the result in part (i).

(Hint: You will need to do a change of order of integration in a double integral.)

Part (i) I differentiated $\displaystyle x(t)$ twice and got the result by applying Leibniz rule and the fundamental theorem of calculus.

For part (ii) I don't even know where to start. Where would I be using a double integral? I tried proving it just by integrating $\displaystyle x''(t)$ but I wasn't sure if i could do that.

Any help would be appreciated.

Thanks.