Variable separable form of a first order ODE example 1

Introduction : In this solution, Variable separable form of a first order ODE example 1 is provided with step by step explanation.

Question: Solve

\[ \frac{dy}{dx} = \frac{1+y^2}{1+x^2}\]

Solution:

Given ODE

\[ \frac{dy}{dx} = \frac{1+y^2}{1+x^2}\]

We can write it as f(y)dy=g(x)dx

\[\frac{1}{1+y^2}dy =\frac{1+y^2}{1+x^2} dx\]

this is called variable separable form of an ODE

Now integrate above equation

\[tan^{-1}(y) = tan^{-1}(x)+tan^{-1}(c)\]

\[tan^{-1}(y) = tan^{-1}(\frac{x+c}{1-xc})\]

Cancellation of tan^(-1) implies

\[y=\frac{x+c}{1-xc} \]

this is the solution of the given ordinary differential equation.