How to solve the differential equation $\frac{d^2 y}{dx^2}=ay^{-2}$

Question

How to solve the following differential equation to get $y$ as a function of $x$: $$\frac{d^2y}{dx^2}=\frac{a}{y^2}$$

Answer 1

HINT:

Abbreviating $y(x)=y,$

$$y'' = \frac{a}{y^2}\tag1$$

Multiply both sides by $ 2 y^{'}$ and integrate

$$y'^2 = \frac{-2a}{y} +c $$

$$ x= \int \frac {\sqrt y dy}{\sqrt{ c y -2 a }} \tag2 $$

Can you continue?

Answer 2

Usually in a DE course, as part of the chapter on 1st-order equations, two special cases of 2nd-order equations are mentioned: 1. No dependent variable and 2. No independent variable. This equations is the second kind. The trick is to let $w=y'$ and then express $y''$ in terms of $\frac{dw}{dy}$ so that $y$ plays the role of the independent variable.

$$y'' = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{dw}{dx} = \frac{dw}{dy}\frac{dy}{dx} = \frac{dw}{dy}w.$$

Your equation becomes

$$w\frac{dw}{dy} = \frac{a}{y^2}$$

which is 1st-order separable. Once you know $w=y' =$ crud, integrate to get $y$.