Consider the following system of linear differential equations in $y_1(t)$ and $y_2(t)$:
\begin{align}
y'_1(t)=a_1(t)+a_2(t)y_1(t)+a_3(t)y_2(t),\\
y'_2(t)=b_1(t)+b_2(t)y_1(t)+b_3(t)y_2(t),
\end{align}
together with two boundary conditions, say $y_1(0)=y_1^*$ and $y_2(0)=y_2^*$.
If I am not mistaken, this is a system of linear, non-autonomous ordinary differential equations. I know how to solve this if the coefficients $a$ and $b$ are independent of time, and also how to solve a single non-autonomous ODE (so if $a_3=0$ or $b_2=0$). However, online and in my textbooks I cannot find how to solve the system of ODE's if the coefficients are time-varying.
Does anyone have an idea how to proceed, or references? Thanks in advance!
Best Answer
In general there is no "closed-form" solution. The homogeneous case ($a_1 = b_1 = 0$) can be written in terms of a "time-ordered exponential". Solutions of the non-homogeneous case can then be found as integrals involving the time-ordered exponential.