Question

Solve the differential equation: $(1+x^2)dy=(1+y^2)dx$.

Answer 1

Give differential equation is $(1+x^2)dy=(1+y^2)dx$

$\Rightarrow \int \frac{dy}{1+y^2}=\int \frac{dx}{1+x^2}$

Integrating both the sides $\int \frac{dy}{1+y^2}=\int \frac{dx}{1+x^2}$

${\tan }^{-1}y={\tan }^{-1}x+{\tan }^{-1}C$

$\Rightarrow {\tan }^{-1}y-{\tan }^{-1}x={\tan }^{-1}C$

${\tan }^{-1}\left(\frac{y-x}{1+xy}\right)={\tan }^{-1}C$

$\Rightarrow \frac{y-x}{1+xy}=C$