Internal problem ID [12569]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {x^{\prime \prime }+x+x^{3}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 56
dsolve(diff(x(t),t$2)+x(t)+x(t)^3=0,x(t), singsol=all)
\[ x \left (t \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (\sqrt {3}\, \sqrt {2}\, t +2 c_{1} \right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}}}{2}, \frac {i c_{2} \sqrt {3}}{3}\right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}} \]
✓ Solution by Mathematica
Time used: 60.261 (sec). Leaf size: 169
DSolve[x''[t]+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*}