66.8.3 problem 4

Internal problem ID [16054]
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 : Thursday, October 02, 2025 at 10:40:01 AM
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: 0.162 (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: 0.334 (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)
 

\[ t + \frac {\left (3 \cos ^{2}{\left (y{\left (t \right )} \right )} - 5\right ) \cos {\left (y{\left (t \right )} \right )}}{8 \left (\cos ^{4}{\left (y{\left (t \right )} \right )} - 2 \cos ^{2}{\left (y{\left (t \right )} \right )} + 1\right )} + \frac {3 \log {\left (\cos {\left (y{\left (t \right )} \right )} - 1 \right )}}{16} - \frac {3 \log {\left (\cos {\left (y{\left (t \right )} \right )} + 1 \right )}}{16} = C_{1} \]