2022-01-04

Chebyshev's differential equation has the form (1 - x ^ 2) * (d ^ 2 * y)/(d * x ^ 2) - x * (dy)/(dx) + v ^ 2 * y = 0 Find the possible regular singular points of the differential equation.

alenahelenash

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

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