Internal problem ID [9806]
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]
\[ \boxed {y^{\prime }-y^{2}-m y \cot \left (x \right )-b^{2} \sin \left (x \right )^{2 m}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 301
dsolve(diff(y(x),x)=y(x)^2+m*y(x)*cot(x)+b^2*sin(x)^(2*m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m -2}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}+1} b \left (\left (-m -2\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right )^{4}+\left (\left (-m -2\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )+3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) \cot \left (x \right )^{2}+3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) \left (-c_{1} \sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m -2}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}+1} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) b \right )+\cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m -2}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}+1} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) b \right )\right )}{3 c_{1} \cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m -2}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}+1} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) b \right )+3 \sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m -2}}\, \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}+1} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \cot \left (x \right ) b \right )} \]
✓ Solution by Mathematica
Time used: 1.5 (sec). Leaf size: 233
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 \frac {b \sqrt {\cos ^2(x)} \sec (x) \sin ^m(x) \left (-\cos \left (\frac {b \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}\right )+c_1 \sin \left (\frac {b \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}\right )\right )}{\sin \left (\frac {b \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}\right )+c_1 \cos \left (\frac {b \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}\right )} \\ y(x)\to b \sqrt {\cos ^2(x)} \sec (x) \sin ^m(x) \tan \left (\frac {b \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}\right ) \\ \end{align*}