# Solve differential equation xy'=(1-y^2)^{frac{1}{2}}

Question
Solve differential equation $$xy'=(1-y^2)^{\frac{1}{2}}$$

2020-11-09
The given differential equation is $$\sqrt{1-y^2}$$
$$\Rightarrow x \frac{dy}{dx}= \sqrt{1-y^2}$$
$$\Rightarrow \frac{1}{\sqrt{1-y^2}} dy= \frac{dx}{x}$$
Integrating both sides, we obtain
$$\int \frac{1}{\sqrt{1-y^2}} dy=\int \frac{dx}{x}$$
$$\Rightarrow sin^{-1}(y)= ln (|x|)+c$$
$$\Rightarrow y= \sin (ln(|x|)+c)$$

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