Internal problem ID [13866]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\frac {\left (5-2 \cos \left (x \right )\right )^{3} \sin \left (x \right ) \cos \left (y\right )^{4}}{\sin \left (y\right )}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 230
dsolve(diff(y(x),x)=((5-2*cos(x))^3*sin(x)*cos(y(x))^4)/sin(y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \pi -\arccos \left (\frac {2 \,9^{\frac {1}{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 )^{\frac {1}{3}}}{24 c_{1} -2793-6 \cos \left (4 x \right )-924 \cos \left (2 x \right )+120 \cos \left (3 x \right )+3360 \cos \left (x \right )}\right ) y \left (x \right ) = \pi -\arccos \left (\frac {\left (-1+i \sqrt {3}\right ) 9^{\frac {1}{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 )^{\frac {1}{3}}}{24 c_{1} -2793-6 \cos \left (4 x \right )-924 \cos \left (2 x \right )+120 \cos \left (3 x \right )+3360 \cos \left (x \right )}\right ) y \left (x \right ) = \arccos \left (\frac {\left (1+i \sqrt {3}\right ) 9^{\frac {1}{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 )^{\frac {1}{3}}}{24 c_{1} -2793-6 \cos \left (4 x \right )-924 \cos \left (2 x \right )+120 \cos \left (3 x \right )+3360 \cos \left (x \right )}\right ) \end{align*}
✓ Solution by Mathematica
Time used: 9.673 (sec). Leaf size: 311
DSolve[y'[x]==((5-2*Cos[x])^3*Sin[x]*Cos[y[x]]^4)/Sin[y[x]],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*}