#### 2.547   ODE No. 547

$y'(x)^4-4 y(x) \left (x y'(x)-2 y(x)\right )^2=0$ Mathematica : cpu = 2.44581 (sec), leaf count = 490

$\left \{\text {Solve}\left [\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}+x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}-\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{4 \sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)}}+\frac {1}{4} \log (y(x))=c_1,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )-\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}+x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}=c_1,y(x)\right ],\text {Solve}\left [\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}+x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}+\frac {1}{4} \left (\log (y(x))-\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {1}{4} \left (\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )-\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}+x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.192 (sec), leaf count = 118

$\left \{ {1 \left ( \sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}-x \right ) ^{{1\sqrt {{x}^{2}y \left ( x \right ) -4\, \left ( y \left ( x \right ) \right ) ^{3/2}}{\frac {1}{\sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}}}{\frac {1}{\sqrt {y \left ( x \right ) }}}}}{\frac {1}{\sqrt {y \left ( x \right ) }}} \left ( \left ( \sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}+x \right ) ^{{1\sqrt {{x}^{2}y \left ( x \right ) -4\, \left ( y \left ( x \right ) \right ) ^{3/2}}{\frac {1}{\sqrt {{x}^{2}-4\,\sqrt {y \left ( x \right ) }}}}{\frac {1}{\sqrt {y \left ( x \right ) }}}}} \right ) ^{-1}}-{\it \_C1}=0,y \left ( x \right ) ={\frac {{x}^{4}}{16}} \right \}$