By using a suitable substitution, bring the equation…

By using a suitable substitution, bring the equation $(a^2+y^2)dx+2x\sqrt{ax-x^2}\,dy=0$ to a form with separated variables, and then integrate it, given $y(a)=0$.

Step 1: Separate variables

\[\frac{dx}{2x\sqrt{ax-x^2}}=-\frac{dy}{a^2+y^2}.\]

Step 2: Integrate with $x=a\sin^2 heta$

With $x=a\sin^2\theta$, $\sqrt{ax-x^2}=a\sin\theta\cos\theta$, and the left integral reduces to $\dfrac1a\displaystyle\int\csc^2\theta\,d\theta=-\dfrac1a\cot\theta=-\dfrac1a\sqrt{\dfrac{a-x}{x}}$. The right side gives $-\dfrac1a\arctan\dfrac{y}{a}$. Equating and clearing $-1/a$:

\[\sqrt{\frac{a-x}{x}}=\arctan\frac{y}{a}+C.\]

Step 3: Apply the initial condition

With $y(a)=0$: $\sqrt{0}=\arctan0+C$, so $C=0$, giving

\[\boxed{\,\sqrt{\dfrac{a-x}{x}}=\arctan\dfrac{y}{a}.}\]