Question
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$
✓
Answer
We have,
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$
$\Rightarrow\frac{1}{(1+\text{y}^2)}\text{dy}=(1+\text{x}^2)\text{dx}$
Integrating both sides, we get
$\int\frac{1}{(1+\text{y}^2)}\text{dy}=\int(1+\text{x}^2)\text{dx}$
$\Rightarrow\tan^{-1}\text{y = x}+\frac{\text{x}^3}{3}+\text{C}$
Hence, $\tan^{-1}\text{y = x}+\frac{\text{x}^3}{3}+\text{C}$ is the required solution.
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$
$\Rightarrow\frac{1}{(1+\text{y}^2)}\text{dy}=(1+\text{x}^2)\text{dx}$
Integrating both sides, we get
$\int\frac{1}{(1+\text{y}^2)}\text{dy}=\int(1+\text{x}^2)\text{dx}$
$\Rightarrow\tan^{-1}\text{y = x}+\frac{\text{x}^3}{3}+\text{C}$
Hence, $\tan^{-1}\text{y = x}+\frac{\text{x}^3}{3}+\text{C}$ is the required solution.
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