It's about a simple recurrent ODE arising from a perturbation expansion $$ \ddot y_0 + y_0 = 0, \\ \ddot y_n + y_n = - \dot y_{n-1}, n \ge 1 $$ Now the first equation is easily solved with the ansatz $ y = e^{rt}$, resulting in $y_0 = A e^{it} + c.c.$. Theoretically the next solutions should be straightforward as the homogenous equation is the same and we can find the particular solution by variation of parameters, but the calculation seems a little (too) messy considering that the paper I'm reading just mentions the next solutions as $y_1 = - \frac{1}{2} A t e^{it} + c.c. $ and $y_2 = \frac{1}{8} A (t^2 - t)e^{it} + c.c. $ without further calculations or comments.
It feels like there's a faster way or a trick that I'm not aware of. Further, does the fact that the second solution is always equal to the complex conjugate simplify the calculation?