Coupled differential equation with linear coefficients

Question

I am looking to solve the following set of coupled differential equation: \begin{align} \frac{d A_1(\xi)}{d\xi} &= C (\xi + B) A_2(\xi), \\ \frac{d A_2(\xi)}{d\xi} &= C (\xi - B) A_1(\xi), \end{align} where $C$ and $B$ are constants with respect to $\xi$. It turns out that this system is easily solved when $B = 0$, but does anyone have an idea of an approach when $B \not= 0$?

Answer 1

Differentiate the first equation wrt $\xi$ $$ \frac{d^2 A_1(\xi)}{d\xi^2} = C A_2(\xi) + C (\xi + B) \frac{dA_2(\xi)}{d\xi} $$ and then substitute in for $A_2$ $$ \frac{d^2 A_1(\xi)}{d\xi^2} = \frac{1 } {\xi+B} \frac{d A_1(\xi)}{d\xi}+ C (\xi + B) C (\xi - B) A_1(\xi). $$ You now have a single equation to solve.