For this problem you may want to recall that if $y=\sinh(x),$ then $\displaystyle{\frac{dy}{dx}=\cosh(x)},$ and $\displaystyle{\frac{d^2y}{dx^2}=\sinh(x)}$.

It can be shown that $y_1=\sinh(2 x),\ y_2=\cosh(2 x)$ and $y_3=e^{2 x}$ are solutions to the differential equation $x D^3y - 9 D^2 y -4 x Dy + 36 y =0$ on $(0,\infty)$.

What does the Wronskian of $y_1, y_2, y_3$ equal on $(0,\infty)$?

$W(y_1, y_2, y_3 )$ = on $(0,\infty)$.

Yes No  1. Is $\{y_1,y_2,y_3\}$ a fundamental set for $x D^3y - 9 D^2 y -4 x Dy + 36 y =0$ on $(0,\infty)$?