For each of the differential equations in find the general soluti… - Vidyadip

Question

For each of the differential equations in find the general solution:
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$

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Answer

Given: Differential equation $\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$
$\Rightarrow\ \text{dy}=\big(1+\text{x}^2\big) \big(1+\text{y}^2\big)\text{dx}\ \Rightarrow \ \frac{\text{dy}}{1+\text{y}^2}$ $=\big(1+\text{x}^2\big)\text{dx}\ \ [\text{Separating variables}]$
Integrating both sides, $\int \frac{1}{\text{y}^2+1}\text{dy}=\int\big(\text{x}^2+1\big)\text{dx}$
$\Rightarrow \ \ \ \ \tan^{-1}\text{y}=\frac{\text{x}^3}{3}+\text{x+c}$

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