Higher-order linear ODEs

Find the general solution of $y^{(5)}-y^{(4)}=0$.

Suppose that the characteristic equation of a third-order differential equation has roots $\pm 2i$ and 3.

1. What is the characteristic equation?
2. Find the corresponding differential equation.
3. Find the general solution.

Solve $1001y'''+3.2y''+\pi y'-\sqrt{4} y = 0$, $y(0)=0$, $y'(0) = 0$, $y''(0) = 0$.

Are $e^{x}$, $e^{x+1}$, $e^{2x}$, $\sin(x)$ linearly independent? If so, show it, if not find a linear combination that gives $0$.

Are $\sin(x)$, $x$, $x\sin(x)$ linearly independent? If so, show it, if not find a linear combination that gives $0$.

Find an equation such that $y=\cos(x)$, $y=\sin(x)$, $y=e^x$ are solutions.