Internal problem ID [9374]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1796 (book 6.205).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{2} y^{\prime \prime }-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 793
dsolve(x*y(x)^2*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{1} x \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\ y \relax (x ) = \frac {c_{1} x \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\ y \relax (x ) = \frac {c_{1} x \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x +2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\ y \relax (x ) = \frac {c_{1} x \left (81 a^{2} c_{1}^{2}+18 a c_{1} {\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x c_{1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} x -2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.206 (sec). Leaf size: 116
DSolve[-a + x*y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {a \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c_1} \left (\frac {y(x)}{x}+\frac {a}{2 c_1}\right )}{\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}\right )}{2 \sqrt {2} c_1{}^{3/2}}-\frac {\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}{2 c_1}-\frac {1}{x}-c_2=0,y(x)\right ] \]