System of differential equations depending on parameter

Question

This is the first time to get a question like this:

How to solve the system $y'=3by+(1-2b)z$, $z'=by+z+e^{4x}$, where $b\in\mathbb{R}$? Any help is welcome.

Answer 1

Hint:

Substracting both differential equations gives us: $$(y-z)'-2b(y-z)=-e^{4x}$$ This DE can easily be solved. It's a first linear DE.