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?
$dX/dt=a \times X(t) + b \times Y(t)$
$dY/dt=c \times Y(t) + d \times X(t)$

Rewriting the above as:
$$\begin{bmatrix} X'\\Y'\end{bmatrix}=\begin{bmatrix}a & b \\ d & c\end{bmatrix}\begin{bmatrix}X\\Y\end{bmatrix}$$
Let's assume for the moment that the determinant of the above matrix is nonzero. Then the matrix has two eigenvalues, $r_1$ and $r_2$, with two corresponding eigenvectors, $\vec{v_1}=(v_{1x},v_{1y})$ and $\vec{v_2}=(v_{2x},v_{2y})$. Then the solutions are given by:
$X=C_1v_{1x}e^{r_1t}+C_2v_{2x}e^{r_2t}$
$Y=C_1v_{1y}e^{r_1t}+C_2v_{2y}e^{r_2t}$
where the constants $C_1$ and $C_2$ are determined by initial conditions.
If the determinant of the above square matrix is zero, obtain the single eigenvalue $r$ and the eigenvector $\vec{v}=(v_x,v_y)$, along with the generalized eigenvector $\vec{w}=(w_x,w_y)$. Then the solutions are given by:
$X=C_1v_{x}e^{r_1t}+C_2w_{x}te^{r_2t}$
$Y=C_1v_{y}e^{r_1t}+C_2w_{y}te^{r_2t}$
