2.113   ODE No. 113

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

\[ a \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] Mathematica : cpu = 0.023155 (sec), leaf count = 16

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

Maple : cpu = 0.032 (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; 
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