Linear ODE where coefficients are functions of $t$

Question

Consider the $2$D dynamical system $$x'=a(t)x+b(t)y$$ $$y'=b(t)x+a(t)y,$$ where $a,b : \mathbb{R} \rightarrow \mathbb{R}$ are given continuous functions.
Is there a way to solve this system explicitly in terms of $a$ and $b$?

Answer 1

Time-dependent systems in general are difficult to handle, but in this particular case you can let $u(t)=x(t)+y(t)$ and $v(t)=x(t)-y(t)$ and get uncoupled separable ODEs for $u$ and $v$.