2.89   ODE No. 89

1. Problem in Latex
2. Mathematica input
3. Maple input

$$x{y}^{\prime }(x)-\sqrt{a^2-x^2}=0$$ ✓ Mathematica : cpu = 0.0119913 (sec), leaf count = 42

$$\left\{\{y(x)\to \sqrt{a^2-x^2}-a{\tanh }^{-1}\left(\frac{\sqrt{a^2-x^2}}{a}\right)+c_1\}\right\}$$ ✓ Maple : cpu = 0.02 (sec), leaf count = 56

$$\{y\left(x\right)=\sqrt{a^2-x^2}-a^2\ln \left(\frac{1}{x}(2a^2+2\sqrt{a^2}\sqrt{a^2-x^2})\right)\frac{1}{\sqrt{a^2}}+\mathit{\_}\mathit{C}\mathit{1}\}$$

Hand solution

$$x{y}^{\prime }=\pm \sqrt{a^2-x^2}$$

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

$$\begin{array}{rl}\int \frac{\sqrt{a^2-x^2}}{x}dx& =\int \frac{\sqrt{a^2-a^2{\sin }^{2}u}}{a\sin u}a\cos \left(u\right)du\\ & =\int \frac{a\sqrt{1-{\sin }^{2}u}}{a\sin u}a\cos \left(u\right)du\\ & =a\int \frac{\cos u}{\sin u}\cos \left(u\right)du\\ & =a\int \frac{{\cos }^{2}u}{\sin u}du\\ & =a\int \frac{1-{\sin }^{2}u}{\sin u}du\\ & =a(\int \frac{1}{\sin u}du-\int \sin udu)\\ \text{(1)}& & =a(\int \frac{1}{\sin u}du+\cos u)\end{array}$$

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

$$\begin{array}{rl}\int \frac{1}{\sin u}du& =\int \frac{1+t^2}{2t}\frac{2}{1+t^2}dt\\ & =\int \frac{1}{t}dt\\ & =\ln \left(t\right)\end{array}$$

Hence $\int \frac{1}{\sin u}du=\ln (\tan \left(\frac{u}{2}\right))$ and from (1)

$$\begin{array}{rl}\int \frac{\sqrt{a^2-x^2}}{x}dx& =a(\int \frac{1}{\sin u}du+\cos u)\\ & =a(\ln (\tan \left(\frac{u}{2}\right))+\cos u)\end{array}$$

But $x=a\sin u$, hence $u=\arcsin \left(\frac{x}{a}\right)$ and the integral becomes

$$\int \frac{\sqrt{a^2-x^2}}{x}dx=a[\ln (\tan \left(\frac{\arcsin \left(\frac{x}{a}\right)}{2}\right))+\cos (\arcsin \left(\frac{x}{a}\right))]$$

Hence the solution is

$$y=\pm a[\ln (\tan \left(\frac{\arcsin \left(\frac{x}{a}\right)}{2}\right))+\cos (\arcsin \left(\frac{x}{a}\right))]+C$$

Maple do not verify the above, but I do not see what is wrong with the solution. Will investigate more later.