So I'm taking a complex analysis course on EdX, and this is the first problem on the first problem set. It's far overdue, so I'm not looking to submit it. I just want to know what are the inner workings of how one gets from the question to the answer.
The integral is: $$K(\lambda) = \int_{-\infty}^{\infty}e^{-\lambda t^2}\frac{dt}{t-ib}$$
The equation that it apparently satisfies is (and I don't know this to start with, this is just from clicking on Show Answer; there are blanks in place of the $'$, $b^2$, and $\displaystyle \frac {b}{\sqrt{\lambda}}$):
$$K' - b^2K=-i\sqrt{\pi}\frac{b}{\sqrt{\lambda}}$$
This is a differentiation w.r.t. parameter problem, so $\displaystyle K'\equiv\frac{dK}{d\lambda}$.
I've tried expanding the $\displaystyle \frac{1}{t-ib}$ term, but that doesn't seem to lead anywhere. I've tried differentiating the $e^{-\lambda t^2}$ term w.r.t. parameter, but that doesn't make the expression look like anything nice either.
I don't know how to go about this, and this is quite literally the first problem of a course in which I can follow the lectures.