# The equations these: dx/dt=−3x−5y, dy/dt=5x+3y What method for solving this system?

The equations these:
$\begin{array}{rl}\frac{\mathrm{d}x}{\mathrm{d}t}& =-3x-5y\\ \frac{\mathrm{d}y}{\mathrm{d}t}& =5x+3y\end{array}$
What method for solving this system?
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driogairea1
Consider the following IVP problem:
$\begin{array}{rl}\frac{\mathrm{d}x}{\mathrm{d}t}& =-3x-5y\\ \frac{\mathrm{d}y}{\mathrm{d}t}& =5x+3y\end{array}$
with $x\left(0\right)={x}_{0}$ and $y\left(0\right)={y}_{0}$.
Then, Laplace-transform both sides of both equations to get:
$\begin{array}{rl}sX\left(s\right)-{x}_{0}& =-3X\left(s\right)-5Y\left(s\right)\\ sY\left(s\right)-{y}_{0}& =5X\left(s\right)+3Y\left(s\right),\end{array}$
which is an algebraic system for $X\left(s\right)={\mathcal{L}}_{s}x\left(t\right)$ and $Y\left(s\right)={\mathcal{L}}_{s}y\left(t\right)$. Solve for the unknowns using (for example) Gauss elimination and compute the inverse Laplace transfrom to get the solution.
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benatudq
The equations may be rewritten
$\left[\begin{array}{c}{x}^{\prime }\left(t\right)\\ {y}^{\prime }\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-3& -5\\ 5& 3\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]$
or
${X}^{\prime }\left(t\right)=AX\left(t\right)$
Do you see the analogy with ordinary differential equations?