I'm a bit stuck with how to do this one. So far I have $$G(x,x') = \begin{cases}{A\sin x} & {0<x<t} \\ {B \cos x} & {t<x<\infty}\end{cases}$$ and $$\frac{dG}{dx} = \begin{cases}{A\cos x} & {0<x<t} \\ {-B \sin x} & {t<x<\infty}\end{cases}$$, which when equating for $A$ and $B$ gives me $$G(x,t) = \begin{cases}{-\cos t \sin x} & {0<x<t} \\ {-\sin t \cos x} & {t<x<\infty}\end{cases}$$
I'm not sure where to go from here though. I tried integrating $$y(t) = \int_0^t \cos x \sin t (t \sin t)dt - \int_t^\infty \sin x \cos t (t\ \sin t)dt$$ But the second integral doesn't converge. Am I doing something wrong?