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

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

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

Answers (1)

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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