I have a differential equation that looks like this: $$ \frac{d\alpha(t)}{dt} = [j a - b]\alpha(t) + \sqrt b e^{jct}$$ where I can solve it putting: $$\alpha(t) = \alpha e^{jct}$$ in the above eqn. leading to: $$\alpha(t) = \frac{\sqrt b}{j[a-\omega_l]-b}e^{jct}$$ Now, I want to replace $a$ in the first eqn. with $$a = a_0 + a_1cos(d t)$$. Though even then the first differential eqn. can be solved using integrating factor method leading to solution involving Bessel function, I was wondering what should be the relationship between $a_1,d$ and $d$ so that I can re-write third eqn. as just: $$\alpha(t) = \frac{\sqrt b}{j[a_0 + a_1 cos(dt)-\omega_l]-b}e^{jct}$$ It is usually refererred as Adiabatic condition where the parameter (here, $\alpha(t)$) evolves adiabatically with other variables of the system but I am not sure how to establish the relationship between different variables for it to be true.
