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Ordinary Differential equation Successive Approximations

Taking $\phi(0,\lambda)$ as constant function, where it is equal to 0 which I consider as a first approximation, gives for the second approximation

$\phi_1(t_1,\lambda)= \frac{sin(\sqrt\lambda)t}{\sqrt\lambda}$

Nonetheless, when I try plugging this back into the integral equation I'm given two separable integrals with a q(s) term that I don't know how to integrate for the third approximation, do I just consider it to be q'(s)? Any help would be greatly appreciated.

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    $\begingroup$ Can you also show us the algorithm to generate the approximations? $\endgroup$ – Sean Roberson Oct 7 '18 at 1:52

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