Find the general solution of the differential equation (1+x^2 ) d… - Vidyadip

Question

Find the general solution of the differential equation $\left(1+x^2\right) d y=\left(1+y^2\right) d x$.

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Answer

The given differential equation can be written in the following form:
$
\frac{d y}{d x}=\frac{1+y^2}{1+x^2}
$
$\because 1+y^2, 1+x^2 \neq 0$, Therefore separating the variables, the given differential equation can be written in the following form:
$
\frac{d y}{1+y^2}=\frac{d x}{1+x^2}
$
Integrating
$
\begin{aligned}
\int \frac{d y}{1+y^2} & =\int \frac{d x}{1+x^2} \\
\text { or } \quad \tan ^{-1} y & =\tan ^{-1} x+C
\end{aligned}
$
This is the general solution of the differential equation.

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