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.
[Math] Second order differential equations with substitution help
calculusderivativesordinary differential equations
Best Answer
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 $$ \left(x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} \right) + 9x\frac{dy}{dx} + 20y = \frac{d^2y}{du^2} + 9\frac{dy}{du} + 20y = 0 $$