14.31.5 problem 12

Internal problem ID [2824]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.6 (Qualitative properties of orbits). Page 417
Problem number : 12
Date solved : Monday, January 27, 2025 at 06:18:59 AM
CAS classification : [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} z^{\prime \prime }+z-2 z^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 60.177 (sec). Leaf size: 181

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

\begin{align*} z(t)\to -\frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-4 c_1}+1\right ) (t+c_2){}^2}}{\sqrt {2}}|\frac {1-\sqrt {1-4 c_1}}{\sqrt {1-4 c_1}+1}\right )}{\sqrt {2} \sqrt {\frac {1}{-1+\sqrt {1-4 c_1}}}} \\ z(t)\to \frac {i \text {sn}\left (\frac {\sqrt {\left (\sqrt {1-4 c_1}+1\right ) (t+c_2){}^2}}{\sqrt {2}}|\frac {1-\sqrt {1-4 c_1}}{\sqrt {1-4 c_1}+1}\right )}{\sqrt {2} \sqrt {\frac {1}{-1+\sqrt {1-4 c_1}}}} \\ \end{align*}