2.114   ODE No. 114

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

$$-x\sqrt{x^2+y(x{)}^2}+x{y}^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.0894469 (sec), leaf count = 12

$$\{\{y(x)\to x\sinh (x+c_1)\}\}$$ ✓ Maple : cpu = 2.852 (sec), leaf count = 28

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

Hand solution

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

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

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

Separable.

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

Integrating

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

Since $y=xv$ then

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

Verification

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