Internal problem ID [14186]
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: 239
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 \,3^{\frac {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 )^{\frac {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 \left (x \right ) &= \frac {\pi }{2}-i \operatorname {arcsinh}\left (\frac {\left (\frac {i 3^{\frac {2}{3}}}{3}+3^{\frac {1}{6}}\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 )^{\frac {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 \left (x \right ) &= \frac {\pi }{2}+i \operatorname {arcsinh}\left (\frac {\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}} 3^{\frac {2}{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*}
✓ 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*}