68.4.28 problem 28

Internal problem ID [17208]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 28
Date solved : Thursday, October 02, 2025 at 01:54:01 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\left (5-2 \cos \left (x \right )\right )^{3} \sin \left (x \right ) \cos \left (y\right )^{4}}{\sin \left (y\right )} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 235

ode:=diff(y(x),x) = (5-2*cos(x))^3*sin(x)*cos(y(x))^4/sin(y(x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \pi -\arccos \left (\frac {2 \,3^{{2}/{3}} \left (\left (-8 c_1 +931+2 \cos \left (4 x \right )+308 \cos \left (2 x \right )-40 \cos \left (3 x \right )-1120 \cos \left (x \right )\right )^{2}\right )^{{1}/{3}}}{-48 \cos \left (x \right )^{4}+480 \cos \left (x \right )^{3}-1800 \cos \left (x \right )^{2}+24 c_1 +3000 \cos \left (x \right )-1875}\right ) \\ y &= \frac {\pi }{2}-i \operatorname {arcsinh}\left (\frac {\left (3^{{1}/{6}}+\frac {i 3^{{2}/{3}}}{3}\right ) \left (\left (-8 c_1 +931+2 \cos \left (4 x \right )+308 \cos \left (2 x \right )-40 \cos \left (3 x \right )-1120 \cos \left (x \right )\right )^{2}\right )^{{1}/{3}}}{16 \cos \left (x \right )^{4}-160 \cos \left (x \right )^{3}+600 \cos \left (x \right )^{2}-1000 \cos \left (x \right )-8 c_1 +625}\right ) \\ y &= \frac {\pi }{2}+i \operatorname {arcsinh}\left (\frac {3^{{2}/{3}} \left (\left (-8 c_1 +931+2 \cos \left (4 x \right )+308 \cos \left (2 x \right )-40 \cos \left (3 x \right )-1120 \cos \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (i-\sqrt {3}\right )}{-48 \cos \left (x \right )^{4}+480 \cos \left (x \right )^{3}-1800 \cos \left (x \right )^{2}+24 c_1 +3000 \cos \left (x \right )-1875}\right ) \\ \end{align*}

✓ Mathematica. Time used: 7.631 (sec). Leaf size: 311

ode=D[y[x],x]==((5-2*Cos[x])^3*Sin[x]*Cos[y[x]]^4)/Sin[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sec ^{-1}\left (-\frac {1}{2} \sqrt [3]{-\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to \sec ^{-1}\left (-\frac {1}{2} \sqrt [3]{-\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to -\sec ^{-1}\left (\frac {1}{2} \sqrt [3]{\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to \sec ^{-1}\left (\frac {1}{2} \sqrt [3]{\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to -\sec ^{-1}\left (\frac {1}{2} (-1)^{2/3} \sqrt [3]{\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to \sec ^{-1}\left (\frac {1}{2} (-1)^{2/3} \sqrt [3]{\frac {3}{2}} \sqrt [3]{-2240 \cos (x)+616 \cos (2 x)-80 \cos (3 x)+4 \cos (4 x)+1862+c_1}\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 45.168 (sec). Leaf size: 289

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(5 - 2*cos(x))**3*sin(x)*cos(y(x))**4/sin(y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (\frac {\left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {1}{C_{1} + 2 \cos ^{4}{\left (x \right )} - 20 \cos ^{3}{\left (x \right )} + 75 \cos ^{2}{\left (x \right )} - 125 \cos {\left (x \right )}}}}{6} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (\frac {\left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {1}{C_{1} + 2 \cos ^{4}{\left (x \right )} - 20 \cos ^{3}{\left (x \right )} + 75 \cos ^{2}{\left (x \right )} - 125 \cos {\left (x \right )}}}}{6} \right )} + 2 \pi , \ y{\left (x \right )} = - \operatorname {acos}{\left (\sqrt [3]{\frac {1}{C_{1} + 6 \cos ^{4}{\left (x \right )} - 60 \cos ^{3}{\left (x \right )} + 225 \cos ^{2}{\left (x \right )} - 375 \cos {\left (x \right )}}} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {\left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {1}{C_{1} + 2 \cos ^{4}{\left (x \right )} - 20 \cos ^{3}{\left (x \right )} + 75 \cos ^{2}{\left (x \right )} - 125 \cos {\left (x \right )}}}}{6} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (\frac {\left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {1}{C_{1} + 2 \cos ^{4}{\left (x \right )} - 20 \cos ^{3}{\left (x \right )} + 75 \cos ^{2}{\left (x \right )} - 125 \cos {\left (x \right )}}}}{6} \right )}, \ y{\left (x \right )} = \operatorname {acos}{\left (\sqrt [3]{\frac {1}{C_{1} + 6 \cos ^{4}{\left (x \right )} - 60 \cos ^{3}{\left (x \right )} + 225 \cos ^{2}{\left (x \right )} - 375 \cos {\left (x \right )}}} \right )}\right ] \]