I trying to do this exercise: 
Following the hint, I consider the integral equation $$\phi(t,\lambda) = \frac{\sin(\sqrt{\lambda}t)}{\sqrt{\lambda}} + \frac{1}{\sqrt{\lambda}} \int_0^{t} \sin(\sqrt{\lambda}(t-s))q(s)\phi(s,\lambda)ds$$
For the first term of the successive approximation I say: $\phi_0(t,\lambda) = 0$, because $0$ is our initial data. Then $$\phi_1(t,\lambda) = \frac{\sin(\sqrt{\lambda}t)}{\sqrt{\lambda}} + \frac{1}{\sqrt{\lambda}} \underbrace{\int_0^{t} \sin(\sqrt{\lambda}(t-s)q(s)\cdot 0)ds}_{0}$$
But then $\phi_2(t,\lambda)$ is complicated to compute because the integral is complicated. Is there a nice way to compute the successive aproximation of this integral equation? Am I doing something wrong?