ODE No. 113

$$a\sqrt{x^2+y(x{)}^2}+x{y}^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.117947 (sec), leaf count = 16

DSolve[-y[x] + a*Sqrt[x^2 + y[x]^2] + x*Derivative[1][y][x] == 0,y[x],x]
 

$$\{\{y(x)\to x\sinh (-a\log (x)+c_1)\}\}$$ ✓ Maple : cpu = 0.039 (sec), leaf count = 33

dsolve(x*diff(y(x),x)+a*(y(x)^2+x^2)^(1/2)-y(x) = 0,y(x))
 

$$\frac{x^a\sqrt{y{\left(x\right)}^2+x^2}}{x}+\frac{x^{a}y\left(x\right)}{x}-c_1=0$$

Hand solution

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

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

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

Separable.

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

Integrating

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

Since $y=xv$ then

$$y=x\sinh (C-a\ln x)$$

Verification

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