82.8.32 problem 36-32

Internal problem ID [21903]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-32
Date solved : Thursday, October 02, 2025 at 08:05:52 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }&=4 x^{3}-4 x \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 40

ode:=diff(diff(x(t),t),t) = 4*x(t)^3-4*x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = c_2 \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}\, \operatorname {JacobiSN}\left (\left (\sqrt {2}\, t +c_1 \right ) \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}, c_2\right ) \]

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 150

ode=D[x[t],{t,2}]==4*x[t]^3-4*x[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-\frac {\left (-2+\sqrt {4-2 c_1}\right ) \left (1+\frac {2 x(t)^2}{-2+\sqrt {4-2 c_1}}\right ) \left (1-\frac {2 x(t)^2}{2+\sqrt {4-2 c_1}}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {1}{\sqrt {4-2 c_1}-2}} x(t)\right ),\frac {2-\sqrt {4-2 c_1}}{\sqrt {4-2 c_1}+2}\right ){}^2}{2 \left (2 x(t)^4-4 x(t)^2+c_1\right )}=(t+c_2){}^2,x(t)\right ] \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*x(t)**3 + 4*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out