Second order differential equations with substitution help

Question

Use the substitution $\mathbf{x = e^u}$ to find the general solution of the differential equation $\mathbf{ x^2\frac{d^2y}{dx^2} +10x\frac{dy}{dx} + 20y = 0}$. The only question of this nature that I've ever done involved substituting $\mathbf{x = \sqrt{t}}$ and differentiating using the chain rule. I've no idea how to attempt this. Any help would be appreciated.

Answer 1

I'm not sure why you're confused, since this substitution is no different from what you've already done. Using the chain rule

$$ \frac{dy}{du} = \frac{dy}{dx}\frac{dx}{du} = \frac{dy}{dx}e^u = x\frac{dy}{dx} $$

$$ \frac{d^2y}{du^2} = \frac{d}{dx}\left(\frac{dy}{du}\right)\frac{dx}{du} = x\frac{d}{dx}\left(x\frac{dy}{dx}\right) = x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} $$

The equation becomes $$ \left(x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} \right) + 9x\frac{dy}{dx} + 20y = \frac{d^2y}{du^2} + 9\frac{dy}{du} + 20y = 0 $$