ODE Substitution: Solving Bessel Equation with x=cosθ

[Jerbearrrrrr]

Apparently this is a Bessel equation
$\sin \theta \frac{d^2 y}{d\theta^2} + \cos \theta \frac{dy}{d\theta} + n(n+1)\sin \theta y = 0$

after using x = cos\theta. The problem says use x = cos \theta anyway. A further substitution may be required, but is not alluded to. The variable 'x' is used in the 'target' equation too.

Could someone just verify that the question is right, please? ie, that the substitution will give the following.

It's supposed to end up as
$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - p^2) y = 0$

thanks

I don't remember seeing any qualifications about p (wasn't my question in the first place, just remembering it from a discussion today).

 

[lippyka]

Yes, the substitution does work. Letting x = cosθ and differentiating with respect to θ, you get\frac{d}{d\theta} = -\sin \theta \frac{d}{dx}Substituting this into the original equation gives-\sin \theta \frac{d^2 y}{dx^2} + \cos \theta \left(-\sin \theta \frac{dy}{dx}\right) + n(n+1)\sin \theta y = 0 Simplifying this givesx^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n(n+1)) y = 0