#### 2.113   ODE No. 113

$a \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0$ Mathematica : cpu = 0.0965685 (sec), leaf count = 16

$\{\{y(x)\to x \sinh (-a \log (x)+c_1)\}\}$ Maple : cpu = 0.021 (sec), leaf count = 33

$\left \{ {\frac {{x}^{a}}{x}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}+{\frac {{x}^{a}y \left ( x \right ) }{x}}-{\it \_C1}=0 \right \}$

Hand solution

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

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

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

Separable.

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

Integrating

\begin {align*} \operatorname {arcsinh}\left ( v\right ) & =-a\ln x+C\\ v & =\sinh \left ( C-a\ln x\right ) \end {align*}

Since $$y=xv$$ then

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

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;
0