14.31.1 problem 8

Internal problem ID [2820]
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 : 8
Date solved : Monday, January 27, 2025 at 06:14:16 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 24.614 (sec). Leaf size: 106

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

\begin{align*} z(t)\to \frac {i \sqrt [4]{2} \text {sn}\left (\left .\frac {\sqrt {\sqrt {c_1} (t+c_2){}^2}}{\sqrt [4]{2}}\right |-1\right )}{\sqrt {-\frac {1}{\sqrt {c_1}}}} \\ z(t)\to -\frac {i \sqrt [4]{2} \text {sn}\left (\left .\frac {\sqrt {\sqrt {c_1} (t+c_2){}^2}}{\sqrt [4]{2}}\right |-1\right )}{\sqrt {-\frac {1}{\sqrt {c_1}}}} \\ z(t)\to 0 \\ \end{align*}