## Solution of Second Order Non Linear Differential Equation Example

To solve the differential equation
$y \frac{d^2y}{dx^2} + (\frac{dy}{dx})^2=1$
let
$\frac{dy}{dx}=p$
then
$\frac{d^2y}{dx^2}= \frac{dp}{dx}= \frac{dy}{dx} \frac{dp}{dy}=p \frac{dy}{dx}$
and the equation becomes
$yp \frac{dp}{dy}+p^2=1$
.
Separation of variables gives
$\frac{p}{1-p^2}dp= \frac{1}{y} dy$

$\int \frac{p}{1-p^2}dp= \int \frac{1}{y} dy$

$- \frac{1}{2} ln(1-p^2)= lny+c=$

$ln(1-p^2)=-2lny-2c=ln(\frac{1}{y^2})-2c$

$1-p^2=e^{ln(\frac{1}{y^2})-2c}=e^{-2c} \frac{1}{y^2}$

$p=\sqrt{1-e^{-2c} \frac{1}{y^2}}=\sqrt{y^2-A}{y}$

$p= \frac{dy}{dx}$
so
$\frac{dy}{dx}=\sqrt{y^2-A}{y}$

Separating variables again
$\frac{y}{\sqrt{y^2-A}}dy= dx$
.
Integration gives
$\sqrt{y^2-A}=x+B \rightarrow y^2=(x+B)^2+A \rightarrow y=\sqrt{(x+B)^2+A}$
.