2.112   ODE No. 112

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

$$-\sqrt{x^2+y(x{)}^2}+x{y}^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.0903801 (sec), leaf count = 13

$$\{\{y(x)\to x\sinh (\log (x)+c_1)\}\}$$ ✓ Maple : cpu = 0.033 (sec), leaf count = 27

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

Hand solution

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

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

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

Separable.

$$\frac{dv}{\sqrt{1+v^2}}=\frac{1}{x}dx$$

Integrating

$$\begin{array}{rl}\mathrm{arcsinh}\left(v\right)& =\ln x+C\\ v& =\sinh (\ln x+C)\end{array}$$

Since $y=xv$ then

$$y=x\sinh (\ln x+C)$$

Verification

ode:=x*diff(y(x),x)=sqrt(x^2+y(x)^2)+y(x); 
y0:=x*sinh(ln(x)+_C1); 
odetest(y(x)=y0,ode) assuming x>= 0; 
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