So I'm working on perturbation theory, and I've taken a ode depending on t to a "pde" depending on t1 = t and t2 = et where e is small.

After solving some things I end up with an ode of the form:

y(t2) + y''(t2) = -2i[(A(t2) + A'(t2))e^{it1} - (B(t2) + B'(t2))e^{-it1}]

Where the A and B terms are constants of integration that depend on t2 since we employ two variables to represent the original one. And so employing the method of undetermined coefficients I determine that

y(t2) = F(t2)e^{it1} + G(t2)e^{-it1}.

Evaluating and plugging in yields

F'' + F = -2i[A + A']

G'' + G = 2i[B + B']

So now both sides are odes and I'm not really how to approach this. Intuitionally, second order ODEs are oscillatory and first order ODEs are exponential. The presence of the i on the right hand would seem to turn the ODE into an oscillatory thing, and so I imagine there is a clever way to solve this. However I'm not sure. Thanks in advance!