\[ x y'(x)-\sqrt {a^2-x^2}=0 \] ✓ Mathematica : cpu = 0.0127715 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \sqrt {a^2-x^2}-a \tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )+c_1\right \}\right \}\] ✓ Maple : cpu = 0.022 (sec), leaf count = 56
\[y \relax (x ) = \sqrt {a^{2}-x^{2}}-\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}}{x}\right )}{\sqrt {a^{2}}}+c_{1}\]
Hand solution
\[ xy^{\prime }=\pm \sqrt {a^{2}-x^{2}}\]
This is separable. \(y^{\prime }=\frac {\pm \sqrt {a^{2}-x^{2}}}{x}\) or \(dy=\frac {\pm \sqrt {a^{2}-x^{2}}}{x}dx\). Hence
\[ y=\pm \int \frac {\sqrt {a^{2}-x^{2}}}{x}dx+C \]
Let \(x=a\sin u\), then \(dx=a\cos \relax (u) du\) and the integral becomes
\begin {align} \int \frac {\sqrt {a^{2}-x^{2}}}{x}dx & =\int \frac {\sqrt {a^{2}-a^{2}\sin ^{2}u}}{a\sin u}a\cos \relax (u) du\nonumber \\ & =\int \frac {a\sqrt {1-\sin ^{2}u}}{a\sin u}a\cos \relax (u) du\nonumber \\ & =a\int \frac {\cos u}{\sin u}\cos \relax (u) du\nonumber \\ & =a\int \frac {\cos ^{2}u}{\sin u}du\nonumber \\ & =a\int \frac {1-\sin ^{2}u}{\sin u}du\nonumber \\ & =a\left (\int \frac {1}{\sin u}du-\int \sin udu\right ) \nonumber \\ & =a\left (\int \frac {1}{\sin u}du+\cos u\right ) \tag {1} \end {align}
For \(\int \frac {1}{\sin u}du\), using half tan angle, let \(t=\tan \left ( \frac {u}{2}\right ) ,du=\frac {2}{1+t^{2}}dt,\sin u=\frac {2t}{1+t^{2}}\), therefore
\begin {align*} \int \frac {1}{\sin u}du & =\int \frac {1+t^{2}}{2t}\frac {2}{1+t^{2}}dt\\ & =\int \frac {1}{t}dt\\ & =\ln \relax (t) \end {align*}
Hence \(\int \frac {1}{\sin u}du=\ln \left (\tan \left (\frac {u}{2}\right ) \right ) \) and from (1)
\begin {align*} \int \frac {\sqrt {a^{2}-x^{2}}}{x}dx & =a\left (\int \frac {1}{\sin u}du+\cos u\right ) \\ & =a\left (\ln \left (\tan \left (\frac {u}{2}\right ) \right ) +\cos u\right ) \end {align*}
But \(x=a\sin u\), hence \(u=\arcsin \left (\frac {x}{a}\right ) \) and the integral becomes
\[ \int \frac {\sqrt {a^{2}-x^{2}}}{x}dx=a\left [ \ln \left (\tan \left ( \frac {\arcsin \left (\frac {x}{a}\right ) }{2}\right ) \right ) +\cos \left ( \arcsin \left (\frac {x}{a}\right ) \right ) \right ] \]
Hence the solution is
\[ y=\pm a\left [ \ln \left (\tan \left (\frac {\arcsin \left (\frac {x}{a}\right ) }{2}\right ) \right ) +\cos \left (\arcsin \left (\frac {x}{a}\right ) \right ) \right ] +C \]
Maple do not verify the above, but I do not see what is wrong with the solution. Will investigate more later.