## Solution of Differental Equation By Transformation of Variables

Some differential equation are made easier to solve by transforming variables.
Example
$\frac{dy}{dx}= \frac{x^2+y^2+y}{x}$
.
Let
$tan \theta = \frac{y}{x}$
then
$y=xtan \theta$
.
$\frac{dy}{dx}=tan \theta + xsec^2 \theta \frac{d \theta}{dx}=\frac{x^2+y^2+y}{x} x+x \frac{y^2}{x^2}+ \frac{y}{x}=x + xtan^2 \theta + tan \theta$
.
Then
$xsec^2 \frac{d \theta}{dx}=x+xtan^2 \theta = xsec^2 \theta \rightarrow \frac{d \theta}{dx}=1 \rightarrow \theta +c= x \rightarrow tan^{-1} (y/x)+c=x$
.
Then
$y=xtan(x-c)$
.