Chebyshevs differential

Solve the separable equation ${\displaystyle -x\frac{dy\left(x\right)}{dx}+v^{2}y\left(x\right)+\frac{d(-x^2+1)y\left(x\right)}{x^2}=0:}$

Solve for ${\displaystyle \frac{dy\left(x\right)}{dx}:}$
${\displaystyle \frac{dy\left(x\right)}{dx}=\frac{dy\left(x\right)-d{x}^{2}y\left(x\right)+v^2{x}^{2}y\left(x\right)}{x^3}}$
INTERMEDIATE STEPS:
Solve for ${\displaystyle \frac{dy\left(x\right)}{dx}:}$
${\displaystyle v^{2}y\left(x\right)+\frac{dy\left(x\right)(-x^2+1)}{x^2}-x\frac{dy\left(x\right)}{dx}=0}$
Write the left hand side as a single fraction.
Bring ${\displaystyle -x\frac{dy\left(x\right)}{dx}+v^{2}y\left(x\right)+\frac{d(-x^2+1)y\left(x\right)}{x^2}}$ together using the common denominator $x^2$:
${\displaystyle \frac{dy\left(x\right)-d{x}^{2}y\left(x\right)+v^2{x}^{2}y\left(x\right)-x^3\frac{dy\left(x\right)}{dx}}{x^2}=0}$
Multiply both sides by a constant to simplify the equation.
Multiply both sides by $x^2$:
${\displaystyle dy\left(x\right)-d{x}^{2}y\left(x\right)+v^2{x}^{2}y\left(x\right)-x^3\frac{dy\left(x\right)}{dx}=0}$
Divide both sides by the sign of the leading coefficient of ${\displaystyle -x^3\frac{dy\left(x\right)}{dx}+dy\left(x\right)-d{x}^{2}y\left(x\right)+v^2{x}^{2}y\left(x\right).}$
Multiply both sides by $-1$: