# Solving second order linear homogeneous differential equation

1. May 9, 2012

### the0

1. The problem statement, all variables and given/known data

Find the set of functions from $(-1,1)→ℝ$ which are solutions of:

$(x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y = 0$

2. Relevant equations

3. The attempt at a solution

There is a hint which says to use the change of variable:
$x=cos(θ)$

doing this I get:

$\frac{dx}{dθ} = -sin(θ)$

$\Rightarrow$

$dx = -sin(θ)dθ$

$\Rightarrow$

$(a): \frac{dy}{dx} = \frac{dy}{dθ}\frac{1}{-sin(θ)}$

$(b): \frac{d^{2}y}{dx^{2}} = \frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)}$

Can I do this?!?

If so, substituting everything in gives:

$-sin^{2}(θ)\frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)} + cos(θ)\frac{dy}{dθ}\frac{1}{-sin(θ)} - 4y = 0$

$\Rightarrow$

$y'' + cot(θ)y' + 4y = 0$

Now.... I am not sure.