Exercise 10

Consider the linear system $$\begin{aligned} x' & = x + 3y + (t - 3t^2) \\ y' & = x - y + (2 - t + t^2) \\ x(0) & = 1 \\ y(0) & = -1. \end{aligned}$$

1. Find the general solution of the homogeneous system $$\begin{aligned} x' & = x + 3y \\ y' & = x - y \end{aligned}$$

2. Find a particular solution for $$\begin{aligned} x' & = x + 3y + (t - 3t^2) \\ y' & = x - y + (2 - t + t^2) \end{aligned}$$

3. Find the solution of $$\begin{aligned} x' & = x + 3y + (t - 3t^2) \\ y' & = x - y + (2 - t + t^2) \\ x(0) & = 1 \\ y(0) & = -1. \end{aligned}$$