Question

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

Forms of linear equations

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\displaystyleλ_{1}={1}\Rightarrow \left\{\left[\begin{array}{c}2\\ -1\end{array}\right]\right\},λ_{2}={3}\Rightarrow \left\{\left[\begin{array}{c}3\\ 1\end{array}\right]\right\}$$

2021-06-18

By theorem 6.19 we know that the solution is

$$\displaystyle{y}={c}_{1}{\left({e}^{{λ}{1}{t}}\right)}{u}_{1}+\ldots+{c}_{n}{\left({e}^{{λ}_{n}{t}}\right)}{u}_{n}$$

with $$\lambda_i$$ the eigenvalues of the matrix A and $$u_i$$, the eigenvalues. Thus for this case we then obtain the general solution:

$$\left[\begin{array}{c}y_1\\ y_2\end{array}\right]=y=c_1e^t\left[\begin{array}{c}2\\ -1\end{array}\right]+c_2e^3t\left[\begin{array}{c}3\\ 1\end{array}\right]\\$$

Thus we obtain: $$y_1=2c_1e^t+3c_2e^3t$$

$$y_2=-c_1e^t+c_2e^3t$$