2.115   ODE No. 115

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

$$-x(y(x)-x)\sqrt{x^2+y(x{)}^2}+x{y}^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.128875 (sec), leaf count = 81

$$\left\{\{y(x)\to \frac{x(-2e^{\frac{2c_1+x^2}{\sqrt{2}}}+e^{\sqrt{2}(2c_1+x^2)}-1)}{2e^{\frac{2c_1+x^2}{\sqrt{2}}}+e^{\sqrt{2}(2c_1+x^2)}-1}\}\right\}$$

✓ Maple : cpu = 0.247 (sec), leaf count = 49

$$\{\ln \left(2\frac{x(\sqrt{2{\left(y\left(x\right)\right)}^2+2x^2}+y\left(x\right)+x)}{y\left(x\right)-x}\right)+\frac{\sqrt{2}{x}^2}{2}-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$

Hand solution

$$x{y}^{\prime }=x(y-x)\sqrt{y^2-x^2}+y$$

Let $y=xu$, then $y^{\prime }=u+x{u}^{\prime }$ and the above becomes

$$\begin{array}{rl}x(u+x{u}^{\prime })& =x(xu-x)\sqrt{{\left(xu\right)}^2-x^2}+xu\\ (u+x{u}^{\prime })& =(xu-x)\sqrt{{\left(xu\right)}^2-x^2}+u\\ x{u}^{\prime }& =(xu-x)x\sqrt{u^2-1}\\ u^{\prime }& =x(u-1)\sqrt{u^2-1}\end{array}$$

Separable.

$$\begin{array}{rl}\frac{du}{(u-1)\sqrt{u^2-1}}& =xdx\\ \frac{-u-1}{\sqrt{u^2-1}}& =\frac{x^2}{2}+C\end{array}$$

But $y=xu$, hence

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

Let $\frac{y}{x}=z$

$$\begin{array}{rl}\frac{-z-1}{\sqrt{z^2-1}}& =\frac{x^2}{2}+C\\ -z-1& =\sqrt{z^2-1}(\frac{x^2}{2}+C)\\ {(-z-1)}^2& =(z^2-1){(\frac{x^2}{2}+C)}^2\\ z^2+1+2z& =z^2{(\frac{x^2}{2}+C)}^2-{(\frac{x^2}{2}+C)}^2\\ z^2(1-{(\frac{x^2}{2}+C)}^2)+2z+1+{(\frac{x^2}{2}+C)}^2& =0\end{array}$$

$$

Solving for $z$ (quadratic formula, some conditions apply), one of the solutions is

$$z=\frac{4C{x}^2+4C^2+x^4+4}{4C{x}^2+4C^2+x^4-4}$$

Hence

$$y=x\frac{4C{x}^2+4C^2+x^4+4}{4C{x}^2+4C^2+x^4-4}$$

Need to work on verification. Kamke gives the final solution as

$$y=x\frac{-2C{x}^2+C^2+x^4+4}{-2C{x}^2+C^2+x^4-4}$$

I am not sure where my error now is. Need to look at this again.