# Approximation for Lambert W containing exponential, to help simplify ODE

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.

• Is anything known about $\omega$ and $C$ separately? For instance, is it possible that $C$ is fixed and $\omega$ is arbitrarily large? If you're only considering the case when $\omega$ is fixed and $C$ is small, there is a simple approximation $$i(t) = C_0 (-C \ln C)^{t/A} - B.$$ – Maxim Sep 3 at 0:09

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

$$\frac{\omega \cos (\omega t) - W\left(\frac{e^{\omega \cos (\omega t)}}{C} \right)}{A} = \frac{i'}{i+B}$$

$$\implies \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta + K = \log|i + B|$$

$$i(t) = K\exp\left( \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta \right) - B$$

• Thanks. It would still be nice if there was a way to simplify the solution or original ODE such that the integral is simplified or eliminated as it's not entirely obvious (to me at least) how this solution behaves with respect to variation of the parameter omega in the domain I mentioned. – Bitrex Sep 2 at 20:04
• @Bitrex Letting $C\ll \omega$ wouldn't solve your problem either because then the Lambert W term becomes infinite – Ninad Munshi Sep 2 at 20:08
• This approximation for W in $0 < x < e$ may work for my needs math.stackexchange.com/questions/26682/… – Bitrex Sep 2 at 21:18
• @Bitrex how can that be if you say $C \ll \omega$ ? Or are your considerations different from that? – Ninad Munshi Sep 2 at 21:21
• @Bitrex and here is another plot that disproves your approximation working, which I got by choosing unsymmetric values for the parameters. I don't think an approximation is possible because of the swing given by the cosine. – Ninad Munshi Sep 2 at 23:27