Internal problem ID [12248]
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).
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: 0.016 (sec). Leaf size: 41
dsolve(diff(x(t),t$2)+x(t)-x(t)^3=0,x(t), singsol=all)
\[ x \left (t \right ) = c_{2} \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}\, \operatorname {JacobiSN}\left (\left (\frac {\sqrt {2}\, t}{2}+c_{1} \right ) \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}, c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 60.266 (sec). Leaf size: 171
DSolve[x''[t]+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*}