I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my current understanding and I will appreciate any clarification, even in the form of references.
I have to solve an ode in the form $$ y''(t)+\phi(t)y(t)=j(t). $$ A straightforward approach I can devise is using Green function. So, I solve the equation $$ G''(t)+\phi(t)G(t)=\delta(t) $$ and I would like to write $$ y(t)=y_0(t)+\int G(t,t')j(t')dt'. $$ The problem is that, unless $\phi(t)$ preserves translation symmetry, the above equation cannot hold. This can be seen immediately by observing that one should have $$ G''(t,t')+\phi(t)G(t,t')=\delta(t-t') $$ but this cannot be generally true as instead happens that $$ G''(t-t')+\phi(t-t')G(t-t')=\delta(t-t') $$ and the solution should be written in another way. Is there a general approach to cope with a problem like this?