Internal problem ID [10562]
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: 54.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}+m y \tan \left (x \right )=b^{2} \cos \left (x \right )^{2 m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 346
dsolve(diff(y(x),x)=y(x)^2-m*y(x)*tan(x)+b^2*cos(x)^(2*m),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right ) \sin \left (x \right )^{2}-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right ) b \sqrt {\cos \left (x \right )^{-2+2 m}}\, \cos \left (x \right )^{-m +1} c_{1} \sin \left (b \sqrt {\cos \left (x \right )^{-2+2 m}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )}{3 \left (c_{1} \cos \left (b \sqrt {\cos \left (x \right )^{-2+2 m}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+\sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )\right )}+\frac {b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \cos \left (x \right ) \left (\left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right ) \sin \left (x \right )^{2}-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )}{3 c_{1} \cos \left (b \sqrt {\cos \left (x \right )^{-2+2 m}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 4.179 (sec). Leaf size: 73
DSolve[y'[x]==y[x]^2-m*y[x]*Tan[x]+b^2*Cos[x]^(2*m),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {b^2} \cos ^m(x) \tan \left (-\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(x)\right )}{m+1}+c_1\right ) \]