Analytical solution to linear coupled ODEs? I have this set of equations and I am trying to find $X(t)$ and $Y(t)$ analytically with initial values known such as $X(0)=X_0$ and $Y(0)=Y_0$. How should I approach to solve it?
$dX/dt=a \times X(t) + b \times Y(t)$
$dY/dt=c \times Y(t) + d \times X(t)$

Using brute force.
The system being $$X'=aX+bY \tag 1$$ $$Y'=cY+dX \tag 2$$ Form $(1)$ extract $Y$ $$Y=\frac{X'-aX}b\tag 3$$ Differentiate $$ Y'=\frac{X''-aX'}b\tag 4$$ Plug in $(2)$
$$\frac{X''-aX'}b=c\frac{X'-aX}b+dX \tag 5$$ $$X''-(a+c)X'+(ac-bd)X=0\tag 6$$ Solve $(6)$ for $X$ and go back to $(2)$ to get $Y$.
