2.115   ODE No. 115

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

\[ -x (y(x)-x) \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] Mathematica : cpu = 0.128875 (sec), leaf count = 81

\[\left \{\left \{y(x)\to \frac {x \left (-2 e^{\frac {2 c_1+x^2}{\sqrt {2}}}+e^{\sqrt {2} \left (2 c_1+x^2\right )}-1\right )}{2 e^{\frac {2 c_1+x^2}{\sqrt {2}}}+e^{\sqrt {2} \left (2 c_1+x^2\right )}-1}\right \}\right \}\]

Maple : cpu = 0.247 (sec), leaf count = 49

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

Hand solution

\[ xy^{\prime }=x\left ( y-x\right ) \sqrt {y^{2}-x^{2}}+y \]

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

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

Separable.

\begin {align*} \frac {du}{\left ( u-1\right ) \sqrt {u^{2}-1}} & =xdx\\ \frac {-u-1}{\sqrt {u^{2}-1}} & =\frac {x^{2}}{2}+C \end {align*}

But \(y=xu\), hence

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

Let \(\frac {y}{x}=z\)

\begin {align*} \frac {-z-1}{\sqrt {z^{2}-1}} & =\frac {x^{2}}{2}+C\\ -z-1 & =\sqrt {z^{2}-1}\left ( \frac {x^{2}}{2}+C\right ) \\ \left ( -z-1\right ) ^{2} & =\left ( z^{2}-1\right ) \left ( \frac {x^{2}}{2}+C\right ) ^{2}\\ z^{2}+1+2z & =z^{2}\left ( \frac {x^{2}}{2}+C\right ) ^{2}-\left ( \frac {x^{2}}{2}+C\right ) ^{2}\\ z^{2}\left ( 1-\left ( \frac {x^{2}}{2}+C\right ) ^{2}\right ) +2z+1+\left ( \frac {x^{2}}{2}+C\right ) ^{2} & =0 \end {align*}

\(\allowbreak \)

Solving for \(z\) (quadratic formula, some conditions apply), one of the solutions is

\[ z=\frac {4Cx^{2}+4C^{2}+x^{4}+4}{4Cx^{2}+4C^{2}+x^{4}-4}\]

Hence

\[ y=x\frac {4Cx^{2}+4C^{2}+x^{4}+4}{4Cx^{2}+4C^{2}+x^{4}-4}\]

Need to work on verification. Kamke gives the final solution as

\[ y=x\frac {-2Cx^{2}+C^{2}+x^{4}+4}{-2Cx^{2}+C^{2}+x^{4}-4}\]

I am not sure where my error now is. Need to look at this again.