Internal problem ID [10003]
Internal file name [OUTPUT/8950_Monday_June_06_2022_06_02_24_AM_14377262/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1682 (book 6.91).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {9 x^{2} y^{\prime \prime }+a y^{3}+2 y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN <- 2nd_order JacobiSN successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve(9*x^2*diff(diff(y(x),x),x)+a*y(x)^3+2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (2 c_{1} x^{3}+\sqrt {2}\, \sqrt {x^{\frac {20}{3}} a}\right ) c_{2}}{2 x^{3}}, i\right ) x^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 5.017 (sec). Leaf size: 41
DSolve[2*y[x] + a*y[x]^3 + 9*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \sqrt [3]{x} \text {sn}\left (\left .\left (c_1+\frac {\sqrt {a x^{20/3}}}{\sqrt {2} x^3}\right ) c_2\right |-1\right ) \]