Question

Solve the given system of differential equations by systematic elimination

Answer 1

$\begin{cases}(D^2+2)x-2y=0\\-2x+(D^2+5)y=0\end{cases}$

Solve for $y(t)$ in the first ODE:

$y=\dfrac{(D^2+2)x}2$

Substitute this into the second ODE:

$-2x+(D^2+5)\dfrac{(D^2+2)x}2=0$

$-4x+(D^4+7D^2+10)x=0$

$(D^4+7D^2+6)x=0$

This ODE is linear with characteristic equation

$r^4+7r^2+6=(r^2+6)(r^2+1)=0$

with roots at $r=\pm i\sqrt6$ and $r=\pm i$, so that the characteristic solution $x_c(t)$ is

$\boxed{x(t)=C_1\cos(\sqrt6\,t)+C_2\sin(\sqrt6\,t)+C_3\cos t+C_4\sin t}$

and from here we can solve for $y(t)$:

$y=\dfrac{(D^2+2)x}2$

$\boxed{y(t)=-2C_1\cos(\sqrt6\,t)-2C_2\sin(\sqrt6\,t)+\dfrac{C_3}2\cos t+\dfrac{C_4}2\sin t}$