2.112   ODE No. 112

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

\[ -\sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] Mathematica : cpu = 0.0218069 (sec), leaf count = 13

\[\left \{\left \{y(x)\to x \sinh \left (c_1+\log (x)\right )\right \}\right \}\]

Maple : cpu = 0.06 (sec), leaf count = 27

\[ \left \{ {\frac {1}{{x}^{2}}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}+{\frac {y \left ( x \right ) }{{x}^{2}}}-{\it \_C1}=0 \right \} \]

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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