Internal problem ID [9378]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1799.
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^{3} y^{\prime \prime }-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 70
dsolve(y(x)^3*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {c_{1} \left (c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} x +c_{1}^{2} x^{2}+a \right )}}{c_{1}} \\ y \relax (x ) = -\frac {\sqrt {c_{1} \left (c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} x +c_{1}^{2} x^{2}+a \right )}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.093 (sec). Leaf size: 58
DSolve[-a + y[x]^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\ y(x)\to \frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\ \end{align*}