14.31.2 problem 9
Internal
problem
ID
[2821]
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
:
9
Date
solved
:
Sunday, March 30, 2025 at 12:21:16 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} z^{\prime \prime }+z+z^{5}&=0 \end{align*}
✓ Maple. Time used: 0.021 (sec). Leaf size: 65
ode:=diff(diff(z(t),t),t)+z(t)+z(t)^5 = 0;
dsolve(ode,z(t), singsol=all);
\begin{align*}
-3 \int _{}^{z}\frac {1}{\sqrt {-3 \textit {\_a}^{6}-9 \textit {\_a}^{2}+9 c_1}}d \textit {\_a} -t -c_2 &= 0 \\
3 \int _{}^{z}\frac {1}{\sqrt {-3 \textit {\_a}^{6}-9 \textit {\_a}^{2}+9 c_1}}d \textit {\_a} -t -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 10.516 (sec). Leaf size: 353
ode=D[z[t],{t,2}]+z[t]+z[t]^5==0;
ic={};
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\frac {\left (z(t)^2-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]\right ) \left (z(t)^2-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,2\right ]\right ) \left (z(t)^2-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]\right ) z(t)^2}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ] \left (z(t)^2-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]\right )}}\right ),\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,2\right ] \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ]\right )}\right ){}^2}{\left (-\frac {1}{3} z(t)^6-z(t)^2+c_1\right ) \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,2\right ] \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}-3 c_1\&,3\right ]\right )}=(t+c_2){}^2,z(t)\right ]
\]
✗ Sympy
from sympy import *
t = symbols("t")
z = Function("z")
ode = Eq(z(t)**5 + z(t) + Derivative(z(t), (t, 2)),0)
ics = {}
dsolve(ode,func=z(t),ics=ics)
Timed Out