The solution of differential equation d y d x = 1 + y 2 1 + x 2 is:

Question

The solution of differential equation $\frac{d y}{d x} = \frac{1 + y^{2}}{1 + x^{2}}$ is:

Smart Achievers · Accepted Answer

(2)

Ans. $y - x = k (1 + x y)$

Explanation:

Given that,

$\frac{d y}{d x} = \frac{1 + y^{2}}{1 + x^{2}}$

$\Rightarrow$ $\frac{d y}{1 + y^{2}} = \frac{d x}{1 + x^{2}}$

On integrating both sides, we get

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

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

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

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

$\Rightarrow$ $y - x = \tan C (1 + x y)$

$\Rightarrow$ $y - x = k (1 + x y)$

where, $k = \tan C$

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