2.114   ODE No. 114

$-x \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0$ Mathematica : cpu = 0.0878615 (sec), leaf count = 12

$\{\{y(x)\to x \sinh (c_1+x)\}\}$ Maple : cpu = 2.004 (sec), leaf count = 28

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

Hand solution

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

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

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

Separable.

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

Integrating

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

Since $$y=xv$$ then

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

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;
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