Internal problem ID [9810]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 55.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-m y \cot \relax (x )-b^{2} \left (\sin ^{2 m}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 361
dsolve(diff(y(x),x)=y(x)^2+m*y(x)*cot(x)+b^2*sin(x)^(2*m),y(x), singsol=all)
\[ y \relax (x ) = \frac {b \sqrt {\left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{-m -2}}\, \left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{\frac {m}{2}+1} \left (3 \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right ) \left (\sin ^{4}\relax (x )\right )+\left (\left (-m -2\right ) \hypergeom \left (\left [\frac {3}{2}, \frac {m}{2}+2\right ], \left [\frac {5}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )+3 \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )\right ) \left (\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )+\left (-m -2\right ) \hypergeom \left (\left [\frac {3}{2}, \frac {m}{2}+2\right ], \left [\frac {5}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right ) \left (\cos ^{4}\relax (x )\right )\right ) \left (-c_{1} \sin \left (\frac {b \sqrt {\left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{-m -2}}\, \left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{\frac {m}{2}+1} \cos \relax (x ) \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )}{\sin \relax (x )}\right )+\cos \left (\frac {b \sqrt {\left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{-m -2}}\, \left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{\frac {m}{2}+1} \cos \relax (x ) \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )}{\sin \relax (x )}\right )\right )}{3 \sin \relax (x )^{4} \left (c_{1} \cos \left (\frac {b \sqrt {\left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{-m -2}}\, \left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{\frac {m}{2}+1} \cos \relax (x ) \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )}{\sin \relax (x )}\right )+\sin \left (\frac {b \sqrt {\left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{-m -2}}\, \left (-\frac {2}{-1+\cos \left (2 x \right )}\right )^{\frac {m}{2}+1} \cos \relax (x ) \hypergeom \left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\frac {\cos ^{2}\relax (x )}{\sin \relax (x )^{2}}\right )}{\sin \relax (x )}\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.597 (sec). Leaf size: 72
DSolve[y'[x]==y[x]^2+m*y[x]*Cot[x]+b^2*Sin[x]^(2*m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {b^2} \sin ^m(x) \tan \left (-\sqrt {b^2} \cos (x) \sin ^{m-1}(x) \sin ^2(x)^{\frac {1}{2}-\frac {m}{2}} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3}{2},\cos ^2(x)\right )+c_1\right ) \\ \end{align*}