ODE No. 112

\[ -\sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] Mathematica : cpu = 0.111233 (sec), leaf count = 13

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

\[\{\{y(x)\to x \sinh (\log (x)+c_1)\}\}\] Maple : cpu = 0.045 (sec), leaf count = 27

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

\[\frac {\sqrt {y \left (x \right )^{2}+x^{2}}}{x^{2}}+\frac {y \left (x \right )}{x^{2}}-c_{1} = 0\]

Hand solution

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

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

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

Separable.

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

Integrating

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

Since \(y=xv\) then

\[ y=x\sinh \left ( \ln x+C\right ) \]

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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