Question

# The coefficient matrix for a system of linear differential equations of the form y_1=Ay has the given eigenvalues and eigenspace bases. Find the gener

Forms of linear equations
The coefficient matrix for a system of linear differential equations of the form $$y_1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$

2020-11-03

The general solution is
$$y=c_1y_1+c_2y_2$$
$$\text{for } y_1=e^{at}(\cos(bt)Re(u)-\sin(bt)Im(u))$$
$$y_2=e^{at}(\sin(bt)Re(u)+\cos(bt)Im(u))$$
We have
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$
Then
$$y_1=e^{3t}\bigg(\cos(t)\begin{bmatrix}0\\0\end{bmatrix}-\sin(t)\begin{bmatrix}2\\1\end{bmatrix}\bigg)$$
$$y_2=e^{t}\bigg(\sin(-t)\begin{bmatrix}0\\0\end{bmatrix}+\cos(-t)\begin{bmatrix}-2\\-1\end{bmatrix}\bigg)$$
Hence the general solution is
$$y=c_1y_1+c_2y_2$$
$$=c_1e^{3t}\bigg(\cos(t)\begin{bmatrix}0\\0\end{bmatrix}-\sin(t)\begin{bmatrix}2\\1\end{bmatrix}\bigg)+c_2e^t\bigg(\sin(-t)\begin{bmatrix}0\\0\end{bmatrix}+\cos(-t)\begin{bmatrix}-2\\-1\end{bmatrix}\bigg)$$
The individual functions are
$$y_1=-2c_1e^{3t}(\sin(t)+\cos(-t))$$
$$y_2=-c_2e^{3t}(\sin(t)+\cos(-t))$$