# Consider the system dot(x)=x-y , dot(y)=x+y a) Write the system in matrix form and find the eigenvalues and eigenvectors

Consider the system $\stackrel{˙}{x}=x-y,\stackrel{˙}{y}=x+y$
a) Write the system in matrix form and find the eigenvalues and eigenvectors (Note: they will be complex valued)
b) Write down the general solution for the system of differential equations using only real valued functions. [Hint : use the equality ${e}^{i\omega t}=\mathrm{cos}\left(\omega t\right)+i\mathrm{sin}\left(\omega t\right)$ ]
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Step 1
Given:
$\frac{dx}{dt}=x-y$
$\frac{dy}{dt}=x+y$
To find: (a) = Write system in matrix form , find eigenvalues and eigenvectors.
(b). = General solution.
Step 2.
Solution: a)
$\frac{dx}{dt}=x-y$
$\frac{dy}{dt}=x+y$
$⇒{x}^{\prime }\left(t\right)=Ax\left(t\right)$ , where A is matrix
$⇒\left[\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$
Required matrix form of the system.
Eigenvalues of matrix A :
$⇒|A-xI|=0$
$⇒|\begin{array}{cc}1-x& -1\\ 1& 1-x\end{array}|=0$
$⇒{\left(1-x\right)}^{2}+1=0$
$⇒{x}^{2}-2x+2=0$
$⇒x=\frac{2±\sqrt{4-8}}{2}$
$=x=1±i$
Hence the eigenvalues are in complex from which are 1+i , 1-i
Eigenvector corresponding to 1+i :
$\left[A-\left(1+i\right)I\right]{v}_{1}=0$
$\left[\begin{array}{cc}1-1-i& -1\\ 1& 1-1-i\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$
$⇒\left[\begin{array}{cc}-i& -1\\ 1& -i\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$
${R}_{2}\to i{R}_{2}+{R}_{1}$
$⇒\left[\begin{array}{cc}-i& -1\\ 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$

$⇒-ix-1=0$
$⇒x=\frac{1}{-i}=i$
$K⇒{v}_{1}=\left[\begin{array}{c}i\\ 1\end{array}\right]$
Eigenvector corresponding to 1-i:
$\left[A-\left(1-i\right)I\right]{v}_{2}=0$
$\left[\begin{array}{cc}1-1+i& -1\\ 1& 1-1+i\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$
$⇒\left[\begin{array}{cc}i& -1\\ 1& i\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$
${R}_{2}\to i{R}_{2}-{R}_{1}$
$⇒\left[\begin{array}{cc}i& -1\\ 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=0$