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 : Sunday, March 30, 2025 at 12:21:11 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 20

ode:=diff(diff(z(t),t),t)+z(t)^3 = 0; 
dsolve(ode,z(t), singsol=all);
 

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

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

ode=D[z[t],{t,2}]+z[t]^3==0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 11.920 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(z(t)**3 + Derivative(z(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=z(t),ics=ics)
 

\[ \left [ \frac {z{\left (t \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi } z^{4}{\left (t \right )}}{2 C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} + t, \ \frac {z{\left (t \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi } z^{4}{\left (t \right )}}{2 C_{1}}} \right )}}{4 \sqrt {C_{1}} \Gamma \left (\frac {5}{4}\right )} = C_{2} - t\right ] \]