Internal problem ID [9357]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1778.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{2} y^{\prime \prime }-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.083 (sec). Leaf size: 369
dsolve(y(x)^2*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{1} \left (a^{2} c_{1}^{2}+2 a c_{1} {\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}}{2} \\ y \relax (x ) = \frac {c_{1} \left (a^{2} c_{1}^{2}+2 a c_{1} {\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} a^{2}-2 \textit {\_Z} c_{1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} x \right )}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.178 (sec). Leaf size: 65
DSolve[-a + y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left (\frac {y(x) \sqrt {-\frac {2 a}{y(x)}+c_1}}{c_1}+\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {-\frac {2 a}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(x+c_2){}^2,y(x)\right ] \]