Electrical – Convert varying voltage to resistance

What you are asking is how flat the spectrum of various light measuring devices is. Good devices of either type will have datasheets that tell you this. Usually there is something called a "spectral sensitivity" graph.

If I remember right, photodiodes have narrower spectral sensitivity than CdS cells in general, but can be made to different center wavelengths. For example, 980nm is a common wavelength for IR detectors.

One way of getting a flat spectrum is to use multiple sensors, each peaking at a different wavelength or with a different filter in front of them. Software then uses the various values to compute the "flat equivalent" intensity.

You won't (easily) get a closed form for \$\int e^{p(t)}dt\$.

The solution to the differential equation that you wrote is:

$$i(t)=\left(\frac{V}{L}\right)e^{\frac{-R_{mod}\sin(\omega t)}{L\omega}-\frac{R_{offset}t}{L}}\left(\int_0^t e^{\frac{R_{mod}\sin(\omega \tau)}{L\omega}+\frac{R_{offset}\tau}{L}} d\tau\right)+ Ce^{\frac{-R_{mod}\sin(\omega t)}{L\omega}-\frac{R_{offset}t}{L}}$$ where \$C\$ is an unknown determined by the initial condition. Obviously the term involving integration is the troublesome bit so lets focus on that piece.

The power series for \$e^{x}\$ converges uniformly on bounded sets so we can exchange the power series with the integral giving $$\int_0^t e^{\frac{R_{mod}\sin(\omega \tau)}{L\omega}+\frac{R_{offset}\tau}{L}} d\tau = \sum_{n=0}^{\infty}\frac{1}{n!}\int_0^t\left(\frac{R_{mod}\sin(\omega \tau)}{L \omega}+\frac{R_{offset} \tau}{L} \right)^n d\tau.$$

Integrating the inner terms involves integrating terms of the form \$\tau^k\sin(\omega \tau)^{n-k}\$ which have closed form solutions.

The real upshot is that if we only want approximate solutions we can bound the tail of this series by bounding the integrands and using that $$\sum_{n=N+1}^{\infty}\frac{1}{n!}x^n = e^x-\frac{e^x \Gamma(N+1,x)}{N!}.$$

Then we will have a closed formed approximate solution \$\tilde{i}_N(t)\$ which approximates \$i(t)\$ and we will know a bound for the error \$|\tilde{i}_N(t) - i(t)|\$.