2.538   ODE No. 538

\[ 2 \left (x y'(x)+y(x)\right )^3-y(x) y'(x)=0 \]

✓ Mathematica : cpu = 2.92399 (sec), leaf count = 179

DSolve[-(y[x]*Derivative[1][y][x]) + 2*(y[x] + x*Derivative[1][y][x])^3 == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \frac {\int _1^x\frac {\text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3} \tan ^{-1}\left (\sqrt {8 \text {$\#$1}-1}\right )}{\text {$\#$1} \sqrt {8 \text {$\#$1}-1}}-14 \log \left (\text {$\#$1}^2 (8 \text {$\#$1}-1)\right )+\log \left (\text {$\#$1}^{14} (8 \text {$\#$1}-1)^{15/2} \left (\text {$\#$1}-\sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}\right )\right )+\log \left (\text {$\#$1}^{12} (8 \text {$\#$1}-1)^{13/2} \left (\text {$\#$1}+\sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}\right )\right )+\frac {3 \sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}}{\text {$\#$1}}\& \right ][c_1+2 \log (K[1])]}{K[1]}dK[1]}{x}\right \}\right \}\]

✓ Maple : cpu = 4.116 (sec), leaf count = 1725

dsolve(2*(x*diff(y(x),x)+y(x))^3-y(x)*diff(y(x),x)=0,y(x))
 

\[\int _{\textit {\_b}}^{x}\frac {-6^{{2}/{3}} \left (-9 \textit {\_a}^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} y \left (x \right )^{2}-2 y \left (x \right )}{\textit {\_a}}}}{9}+y \left (x \right )\right ) y \left (x \right )\right )^{{2}/{3}}+6 \textit {\_a} y \left (x \right ) \left (6^{{1}/{3}} \left (-9 \textit {\_a}^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} y \left (x \right )^{2}-2 y \left (x \right )}{\textit {\_a}}}}{9}+y \left (x \right )\right ) y \left (x \right )\right )^{{1}/{3}}-1\right )}{6 \textit {\_a}^{2} y \left (x \right )+\textit {\_a} 6^{{2}/{3}} \left (-9 \textit {\_a}^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} y \left (x \right )^{2}-2 y \left (x \right )}{\textit {\_a}}}}{9}+y \left (x \right )\right ) y \left (x \right )\right )^{{2}/{3}}}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {6 x 6^{{1}/{3}} \left (-9 \textit {\_f} \,x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right )\right )^{{1}/{3}}}{6^{{2}/{3}} \left (-9 \textit {\_f} \,x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right )\right )^{{2}/{3}}+6 \textit {\_f} x}-\left (\int _{\textit {\_b}}^{x}\frac {324 \textit {\_f} \left (\frac {\textit {\_f} \left (\frac {\textit {\_f} \sqrt {3}\, \left (\textit {\_a} \textit {\_f} -\frac {1}{27}\right )}{\sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}-\frac {2 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}{18}+\textit {\_f} \right ) \textit {\_a}}{3}\right ) \textit {\_a}}{\left (-9 \textit {\_f} \,\textit {\_a}^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right )\right )^{{2}/{3}}}-\frac {6^{{2}/{3}} \left (\frac {\textit {\_f} \sqrt {3}\, \left (\textit {\_a} \textit {\_f} -\frac {1}{27}\right )}{\sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}+\frac {\left (-\frac {2 \sqrt {3}\, \sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}}{3}\right )}{6}\right ) 6^{{1}/{3}} \textit {\_a}}{\left (6^{{2}/{3}} \left (-9 \textit {\_f} \,\textit {\_a}^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right )\right )^{{2}/{3}}+6 \textit {\_a} \textit {\_f} \right )^{2}}d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0\]