Solve $\displaystyle\frac{dy}{dx} = \frac{1 + y^{2}}{1 + x^{2}}$.

Solve (dy)/(dx) = frac1 + y²1 + x².

Worked Examples · Example 4

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

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The equation separates cleanly — all $y$ -terms with $dy$ on the left, all $x$ -terms with $dx$ on the right:

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

Integrate both sides, using $\displaystyle\int \frac{dt}{1+t^2} = \tan^{-1}t$ :

$\tan^{-1}y = \tan^{-1}x + c.$ …