70.1.14 problem 2.4 (i)

Internal problem ID [14304]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.4 (i)
Date solved : Tuesday, January 28, 2025 at 06:26:02 AM
CAS classification : [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }+x-x^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 43

dsolve(diff(x(t),t$2)+x(t)-x(t)^3=0,x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.161 (sec). Leaf size: 171

DSolve[D[x[t],{t,2}]+x[t]-x[t]^3==0,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-2 c_1}+1\right ) (t+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}}}} \\ x(t)\to \frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-2 c_1}+1\right ) (t+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*}