72.8.3 problem 4
Internal
problem
ID
[14767]
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
:
Tuesday, January 28, 2025 at 07:14:35 AM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }&=-\sin \left (y\right )^{5} \end{align*}
✓ Solution by Maple
Time used: 0.133 (sec). Leaf size: 190
dsolve(diff(y(t),t)=-sin(y(t))^5,y(t), singsol=all)
\[
y = \arctan \left (\frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_{1} {\mathrm e}^{4 \textit {\_Z}}+24 \,{\mathrm e}^{4 \textit {\_Z}} \textit {\_Z} +64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )}}{{\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_{1} {\mathrm e}^{4 \textit {\_Z}}+24 \,{\mathrm e}^{4 \textit {\_Z}} \textit {\_Z} +64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )}+1}, \frac {-{\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_{1} {\mathrm e}^{4 \textit {\_Z}}+24 \,{\mathrm e}^{4 \textit {\_Z}} \textit {\_Z} +64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )}+1}{{\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{8 \textit {\_Z}}+8 \,{\mathrm e}^{6 \textit {\_Z}}+64 c_{1} {\mathrm e}^{4 \textit {\_Z}}+24 \,{\mathrm e}^{4 \textit {\_Z}} \textit {\_Z} +64 t \,{\mathrm e}^{4 \textit {\_Z}}-8 \,{\mathrm e}^{2 \textit {\_Z}}-1\right )}+1}\right )
\]
✓ Solution by Mathematica
Time used: 0.348 (sec). Leaf size: 48
DSolve[D[y[t],t]==-Sin[y[t]]^5,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*}