Construct the general solution of the system of equations

$$\begin{aligned} &\dot{x}+4 x+3 y=0 \\ &\dot{y}+4 y-3 x=0 \end{aligned}$$

in the form

$$\left(\begin{array}{l} x(t) \\ y(t) \end{array}\right)=\mathbf{x}=\sum_{j=1}^{2} a_{j} \mathbf{x}^{(j)} e^{\lambda_{j} t}$$

and find the eigenvectors $\mathbf{x}^{(j)}$ and eigenvalues $\lambda_{j}$.

Explain what is meant by resonance in a forced system of linear differential equations.

Consider the forced system

$$\begin{aligned} &\dot{x}+4 x+3 y=\sum_{j=1}^{2} p_{j} e^{\lambda_{j} t} \\ &\dot{y}+4 y-3 x=\sum_{j=1}^{2} q_{j} e^{\lambda_{j} t} \end{aligned}$$

Find conditions on $p_{j}$ and $q_{j}(j=1,2)$ such that there is no resonant response to the forcing.