Inhomogeneous system of DE in matrix form I have got the following inhomogeneous systems of differential equations:
$$ \dot x = 2x + y + 3 e ^{t} $$
$$ \dot y = 4x - y $$
I have to write down in matrix form the corresponding homogeneous system of equations.
But I am stuck with the part of $3 e ^{t} $
Does anybody have a pointer?

We can write this as the system
$$X'(t) = AX(t) + g(t) = \begin{pmatrix}2 & 1 \\ 4 & -1\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}+\begin{pmatrix}3 e^t \\ 0\end{pmatrix}$$
You could then solve the homogeneous and non-homogeneous part using many different approaches.
If you are looking for theory and examples, see these notes or notes or notes.
If you solve these, you would get
$$\begin{align} x(t) &= \frac{1}{5} c_1 e^{-2 t}+\frac{4}{5} c_1 e^{3 t}-\frac{1}{5} c_2 e^{-2 t}+\frac{1}{5} c_2 e^{3 t} -e^t \\
y(t) &= -\frac{4}{5} c_1 e^{-2 t}+\frac{4}{5} c_1 e^{3 t}+\frac{4}{5} c_2 e^{-2 t}+\frac{1}{5} c_2 e^{3 t} -2 e^t \end{align}$$
