23.4.106 problem 106

Internal problem ID [6408]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 106
Date solved : Friday, October 03, 2025 at 02:05:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 y+a y^{3}+9 x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 36

ode:=2*y(x)+a*y(x)^3+9*x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,\operatorname {JacobiSN}\left (\frac {\left (2 c_1 \,x^{3}+\sqrt {2}\, \sqrt {x^{{20}/{3}} a}\right ) c_2}{2 x^{3}}, i\right ) x^{{1}/{3}} \]

✓ Mathematica. Time used: 1.815 (sec). Leaf size: 41

ode=2*y[x] + a*y[x]^3 + 9*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)**3 + 9*x**2*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a*y(x)**3 + 9*x**2*Derivative(y(x), (x, 2)) + 2*y(x)