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Differential Equations

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Reducing a System to an Equation: Complex Roots

DIFFEQ-IW52YN

The system

$$\left\{\begin{array}{c}x'=x-5y\\\ y'=2x-y\end{array}\right.$$

can be reduced to a single equation. By the second equation, we have $x=\frac{1}{2}(y'+y)$.Plugging this into the first:

$$\frac{1}{2}(y''+y')=\frac{1}{2}(y'+y)-5y$$

...or:

$$y''+9y=0$$

The general solution for $y$ is given by:

A

$y=C_1\cos(3t)+C_2\sin(3t)$

B

$y=C_1e^{3t}+C_2e^{-3t}$

C

$y=Ce^{-9t}$

D

None of the above