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

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.
$$\displaystyleλ{1}={1}\to{\left\lbrace\begin{array}{cc} {2}&{1}\end{array}\right\rbrace},λ{2}={3}\to{\left\lbrace\begin{array}{cc} {3}&{1}\end{array}\right\rbrace}$$

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By theorem 6.19 we know that the solution is $$y=c1(e^λ1t)u1+...+cn(e^λnt)un$$

wiht λi the eigenvalues of the matrix A and ui, the eigenvalues.

Thus for this case we then obtain the general solution:

$$[y1,y2]=y=c1e^t[2,-1]+c2e^3t[3,1]$$

Thus we obtain:

$$y1=2c1e^t+3c2e^3t$$

$$y2=-c1e^t+c2e^3t$$