Find a real general solution of the following systems. Show the details. y'1=y1+2y2, y'2=½y1+y2

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Let us find the real general solution of the system

\begin{align*} y^\prime_1&=y_1+2y_2\\ y^\prime_2&=\frac{1}{2}\cdot y_1+y_2 \end{align*}

Written in matrix form

\pmb{y^\prime}=\pmb{Ay}

where

\begin{align*}\pmb{A}=\begin{bmatrix} 1 & 2\\ \dfrac{1}{2} & 1 \end{bmatrix}\end{align*}

The characteristic equation is:

\begin{align*} \det (\pmb{A}-\lambda \pmb{I})&=0\Leftrightarrow\\ \begin{vmatrix} 1-\lambda & 2\\ \dfrac{1}{2} & 1-\lambda \end{vmatrix}&=0\Leftrightarrow\\ (1-\lambda)^2-1&=0\Rightarrow \lambda_1=0\,\, , \,\, \lambda_2=2 \end{align*}

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