Internal problem ID [9260]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1681.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} y^{\prime \prime }+a y^{3}+2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 31
dsolve(9*x^2*diff(diff(y(x),x),x)+a*y(x)^3+2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{2} \mathrm {sn}\left (\left (\frac {\sqrt {2}\, \sqrt {x^{\frac {20}{3}} a}}{2 x^{3}}+c_{1}\right ) c_{2}| i\right ) x^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 1.667 (sec). Leaf size: 41
DSolve[2*y[x] + a*y[x]^3 + 9*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \sqrt [3]{x} \text {sn}\left (\left .\left (c_1+\frac {\sqrt {a x^{20/3}}}{\sqrt {2} x^3}\right ) c_2\right |-1\right ) \\ \end{align*}