I have a linear ODE derived from electrical engineering of the form:
$$\omega\cos(\omega t) - W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right) = \frac{Ai'(t)}{i(t)}$$
Where A, C are constants, and W is the Lambert W function.
Wolfram Alpha returns the solution in the form of an integral:
$$i(t) = c_1\exp\int_{1}^{t}\frac{\omega\cos(\omega \zeta) - W\left(\frac{e^{\omega\cos(\omega \zeta)}}{C}\right)}{A}d\zeta$$
However when I insert a third constant term, B, on the RHS as:
$$\omega\cos(\omega t) - W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right) = \frac{Ai'(t)}{i(t) + B}$$
The system seems unable to assist with a solution.
However it seems that when C is small with respect to $\omega$ that
$$W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right)$$
this term seems to behave more-or-less as a periodic sinusoid with an added constant/"DC offset."
I'm hoping for an approximation to the expression involving the W function when constant C is small with respect to omega that could allow for a more practical explicit solution to the previous ODE in that regime.
That still has a solution with the $B$ added, just follow the same steps that got you the first solution. This is a separable equation, so we get
$$\frac{\omega \cos (\omega t) - W\left(\frac{e^{\omega \cos (\omega t)}}{C} \right)}{A} = \frac{i'}{i+B}$$
$$\implies \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta + K = \log|i + B|$$
$$i(t) = K\exp\left( \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta \right) - B$$