83.29.5 problem 5

Internal problem ID [19307]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (C) at page 107
Problem number : 5
Date solved : Monday, March 31, 2025 at 07:06:17 PM
CAS classification : [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x) = y(x)^3-y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.16 (sec). Leaf size: 171

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

\begin{align*} y(x)\to -\frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-2 c_1}+1\right ) (x+c_2){}^2}}{\sqrt {2}}|\frac {1-\sqrt {1-2 c_1}}{\sqrt {1-2 c_1}+1}\right )}{\sqrt {\frac {1}{-1+\sqrt {1-2 c_1}}}} \\ y(x)\to \frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-2 c_1}+1\right ) (x+c_2){}^2}}{\sqrt {2}}|\frac {1-\sqrt {1-2 c_1}}{\sqrt {1-2 c_1}+1}\right )}{\sqrt {\frac {1}{-1+\sqrt {1-2 c_1}}}} \\ \end{align*}

✗ Sympy

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

Timed Out