Solution of Differental Equation By Transformation of 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
\[y=xtan(x-c)\]
. 
