Reassemble a system of differential equations 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 ...--prophetes.ai

Reassemble a system of differential equations 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.

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.