83.35.9 problem 9

Internal problem ID [19429]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:39:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \sin \left (y\right )^{3} y^{\prime \prime }&=\cos \left (y\right ) \end{align*}

✓ Solution by Maple

Time used: 1.726 (sec). Leaf size: 3626

dsolve(sin(y(x))^3*diff(y(x),x$2)=cos(y(x)),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \arctan \left (-\frac {\sqrt {2}\, \sqrt {-2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, \frac {\sqrt {\frac {-4 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-2 c_{1}^{2}+2}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {\sqrt {2}\, \sqrt {-2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, -\frac {\sqrt {\frac {-4 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-2 c_{1}^{2}+2}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (\sqrt {2}\, \sqrt {-2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}, \frac {\sqrt {\frac {-4 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-2 c_{1}^{2}+2}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{c_{1}^{2}-1}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {2}\, \sqrt {-2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, -\frac {\sqrt {\frac {-4 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-2 c_{1}^{2}+2}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {\sqrt {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, -\frac {\sqrt {\frac {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}-1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {\sqrt {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, \frac {\sqrt {\frac {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}-1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}}{2}, -\frac {\sqrt {\frac {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}-1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{2 c_{1}^{2}-2}\right ) \\ y \left (x \right ) &= \arctan \left (\sqrt {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}+1}, \frac {\sqrt {\frac {2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}-c_{1}^{2}+1}{c_{1}^{2}}}\, \left (2 \sqrt {\left (c_{1} -1\right )^{2} \left (c_{1} +1\right )^{2} \left (\left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{4}-8 \sec \left (\frac {c_{2}}{c_{1}}\right )^{2}+8\right ) \tan \left (\frac {x}{c_{1}}\right )^{2}+2 \tan \left (\frac {c_{2}}{c_{1}}\right ) \left (\sec \left (\frac {c_{2}}{c_{1}}\right )^{2}-2\right ) \left (\sec \left (\frac {x}{c_{1}}\right )^{2}-2\right ) \tan \left (\frac {x}{c_{1}}\right )+\sec \left (\frac {x}{c_{1}}\right )^{4} \tan \left (\frac {c_{2}}{c_{1}}\right )^{2}\right ) \sec \left (\frac {c_{2}}{c_{1}}\right )^{4} \sec \left (\frac {x}{c_{1}}\right )^{4}}\, \cos \left (\frac {x}{c_{1}}\right )^{4} \cos \left (\frac {c_{2}}{c_{1}}\right )^{4}+c_{1}^{2}-1\right ) \operatorname {csgn}\left (\sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right )\right ) \sec \left (\frac {2 x +2 c_{2}}{c_{1}}\right ) c_{1}}{c_{1}^{2}-1}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.414 (sec). Leaf size: 86

DSolve[Sin[y[x]]^3*D[y[x],{x,2}]==Cos[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {\csc ^2(y(x)) (c_1 \cos (2 y(x))+2-c_1) \log ^2\left (\sqrt {2} \sqrt {c_1} \cos (y(x))+\sqrt {c_1 \cos (2 y(x))+2-c_1}\right )}{2 c_1 \left (-\csc ^2(y(x))+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]