72.8.3 problem 4
Internal
problem
ID
[14696]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
1.
First-Order
Differential
Equations.
Review
Exercises
for
chapter
1.
page
136
Problem
number
:
4
Date
solved
:
Monday, March 31, 2025 at 12:53:47 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }&=-\sin \left (y\right )^{5} \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 89
ode:=diff(y(t),t) = -sin(y(t))^5;
dsolve(ode,y(t), singsol=all);
\[
y = \arctan \left (\operatorname {sech}\left (\operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_1 \,{\mathrm e}^{4 \textit {\_Z}}+24 \textit {\_Z} \,{\mathrm e}^{4 \textit {\_Z}}+64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )\right ), -\tanh \left (\operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_1 \,{\mathrm e}^{4 \textit {\_Z}}+24 \textit {\_Z} \,{\mathrm e}^{4 \textit {\_Z}}+64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )\right )\right )
\]
✓ Mathematica. Time used: 0.348 (sec). Leaf size: 48
ode=D[y[t],t]==-Sin[y[t]]^5;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{10 \sin (K[1])-5 \sin (3 K[1])+\sin (5 K[1])}dK[1]\&\right ]\left [-\frac {t}{16}+c_1\right ] \\
y(t)\to 0 \\
\end{align*}
✓ Sympy. Time used: 0.495 (sec). Leaf size: 60
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(sin(y(t))**5 + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
t + \frac {\left (3 \cos ^{2}{\left (y{\left (t \right )} \right )} - 5\right ) \cos {\left (y{\left (t \right )} \right )}}{8 \left (\cos ^{4}{\left (y{\left (t \right )} \right )} - 2 \cos ^{2}{\left (y{\left (t \right )} \right )} + 1\right )} + \frac {3 \log {\left (\cos {\left (y{\left (t \right )} \right )} - 1 \right )}}{16} - \frac {3 \log {\left (\cos {\left (y{\left (t \right )} \right )} + 1 \right )}}{16} = C_{1}
\]