Mathematics - Differential Equation Question with Solution | TestHub

The equation can be written as $x^2\frac{xdy-ydx}{x^2}=x\sqrt{1-(y/x{)}^2}dx$.

This simplifies to:

$$\frac{d(y/x)}{\sqrt{1-(y/x{)}^2}}=\frac{dx}{x}$$

Integrating both sides, we get:

$${\sin }^{-1}(y/x)=\log |x|+C$$

Given the initial condition $y(1)=0$, we can find the constant $C$:

$${\sin }^{-1}(0/1)=\log |1|+C$$

$$0=0+C$$

So, $C=0$.

Therefore, the solution is $y=x\sin (\log |x|)$.