70.1.15 problem 2.4 (ii)

Internal problem ID [14305]
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 (ii)
Date solved : Tuesday, January 28, 2025 at 06:26:04 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: 0.045 (sec). Leaf size: 55

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

\[ x \left (t \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, \sqrt {3}\, t +2 c_{1} \right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}}}{2}, \frac {i c_{2} \sqrt {3}}{3}\right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}} \]

✓ Solution by Mathematica

Time used: 60.145 (sec). Leaf size: 169

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

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