Reassemble a system of differential equations

Question

Given the system of equations

$\frac{dx}{dt}=-\frac{3}{2}x-\frac{5}{2}y$,

$\frac{dy}{dt}=\frac{5}{2}x+\frac{3}{2}y$,

how can it be converted back into the second-order ODE? I've tried to do it myself, but got tangled up in the variables.

Answer 1

From the second equation, $$x=\frac25\frac{dy}{dt}-\frac35y\ .$$ Differentiate to find a formula for $\frac{dx}{dt}$, substitute both into the first equation, tidy up.

Answer 2

$$x'=-\frac{3}{2}x-\frac{5}{2}y$$ $$y=-\frac{3}{5}x-\frac{2}{5}x'\tag 1$$ $$y'=-\frac{3}{5}x'-\frac{2}{5}x''\tag 2$$ now substitute the $1$ and $2$ in the second equ to get $$-\frac{3}{5}x'-\frac{2}{5}x''=\frac{5}{2}x+\frac{3}{2}(-\frac{3}{5}x-\frac{2}{5}x')$$