# 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.
$\lambda 1=1\to \left\{\begin{array}{cc}2& 1\end{array}\right\},\lambda 2=3\to \left\{\begin{array}{cc}3& 1\end{array}\right\}$
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By theorem 6.19 we know that the solution is $y=c1\left({e}^{\lambda }1t\right)u1+...+cn\left({e}^{\lambda }nt\right)un$

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

Thus for this case we then obtain the general solution:

$\left[y1,y2\right]=y=c1{e}^{t}\left[2,-1\right]+c2{e}^{3}t\left[3,1\right]$

Thus we obtain:

$y1=2c1{e}^{t}+3c2{e}^{3}t$

$y2=-c1{e}^{t}+c2{e}^{3}t$