Internal problem ID [7930]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 350.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime } \cos \relax (y)-\cos \relax (x ) \left (\sin ^{2}\relax (y)\right )-\sin \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.251 (sec). Leaf size: 270
dsolve(diff(y(x),x)*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \arctan \left (-\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{x} \cos \relax (x )+{\mathrm e}^{x} \sin \relax (x )+2 c_{1}}, \frac {\sqrt {\left (2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1}^{2}+{\mathrm e}^{2 x}\right ) \left (2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_{1}^{2}\right )}}{2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1}^{2}+{\mathrm e}^{2 x}}\right ) \\ y \relax (x ) = \arctan \left (-\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{x} \cos \relax (x )+{\mathrm e}^{x} \sin \relax (x )+2 c_{1}}, -\frac {\sqrt {\left (2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1}^{2}+{\mathrm e}^{2 x}\right ) \left (2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_{1}^{2}\right )}}{2 \,{\mathrm e}^{2 x} \cos \relax (x ) \sin \relax (x )+4 c_{1} \sin \relax (x ) {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \cos \relax (x ) c_{1}+4 c_{1}^{2}+{\mathrm e}^{2 x}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.975 (sec). Leaf size: 56
DSolve[-Sin[y[x]] - Cos[x]*Sin[y[x]]^2 + Cos[y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\csc ^{-1}\left (\frac {1}{2} \left (\sin (x)+\cos (x)+2 c_1 e^{-x}\right )\right ) \\ y(x)\to -\csc ^{-1}\left (\frac {1}{2} \left (\sin (x)+\cos (x)+2 c_1 e^{-x}\right )\right ) \\ y(x)\to 0 \\ \end{align*}