2.1647   ODE No. 1647

\[ y''(x)-a \left (x y'(x)-y(x)\right )^r=0 \] Mathematica : cpu = 0.443676 (sec), leaf count = 91


\[\left \{\left \{y(x)\to x \left (c_2-x \left (2-\frac {a (r-1) x^2}{c_1}\right ){}^{\frac {1}{r-1}} \left (x^{2 r-2} \left (-a (r-1) x^2+2 c_1\right )\right ){}^{\frac {1}{1-r}} \, _2F_1\left (-\frac {1}{2},\frac {1}{r-1};\frac {1}{2};\frac {a (r-1) x^2}{2 c_1}\right )\right )\right \}\right \}\] Maple : cpu = 0.563 (sec), leaf count = 60


\[y \relax (x ) = \left (\int -\frac {\left (x^{2} \left (r -1\right ) a -c_{1}\right ) \left (-\frac {1}{x^{2} \left (r -1\right ) a -c_{1}}\right )^{\frac {r}{r -1}} 2^{\frac {r}{r -1}}}{2 x^{2}}d x +c_{2}\right ) x\]