-307.28.1 Find the general solution to the system of DE

[karush]

$\tiny{28.1}$
2000
Find the general solution to the system of differential equations
\begin{align*}\displaystyle
y'_1&=y_1+5y_2\\
y'_2&=-2y_1+-y_2
\end{align*}
why is there a $+-y_2$ in the given
ok going to take this a step at a time... so..$A=\left[\begin{array}{c}1 & 5 \\ -2 & -1 \end{array}\right]$
then
$\left[\begin{array}{c}1-\lambda & 5 \\ -2 & -1-\lambda \end{array}\right]
=\lambda^2+9$ ?

 

[HallsofIvy]

yes, \(\displaystyle \left|\begin{array}{cc}1- \lambda & 5 \\ -2 & -1- \lambda\end{array}\right|= (1- \lambda)(-1- \lambda)+ 10= -1- \lambda+ \lambda+ \lambda^2+ 10= \lambda^2+ 9= 0\).

Now, what are the values of \(\displaystyle \lambda\)?