#### 2.89   ODE No. 89

$x y'(x)-\sqrt {a^2-x^2}=0$ Mathematica : cpu = 0.0124099 (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 = 0.013 (sec), leaf count = 56

$\left \{ y \left ( x \right ) =\sqrt {{a}^{2}-{x}^{2}}-{{a}^{2}\ln \left ( {\frac {1}{x} \left ( 2\,{a}^{2}+2\,\sqrt {{a}^{2}}\sqrt {{a}^{2}-{x}^{2}} \right ) } \right ) {\frac {1}{\sqrt {{a}^{2}}}}}+{\it \_C1} \right \}$

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 $$dx=a\cos \left ( u\right ) du$$ 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 \left ( u\right ) du\nonumber \\ & =\int \frac {a\sqrt {1-\sin ^{2}u}}{a\sin u}a\cos \left ( u\right ) du\nonumber \\ & =a\int \frac {\cos u}{\sin u}\cos \left ( u\right ) 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 \left ( t\right ) \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.