Internal problem ID [9809]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+m y \tan \relax (x )-b^{2} \left (\cos ^{2 m}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 223
dsolve(diff(y(x),x)=y(x)^2-m*y(x)*tan(x)+b^2*cos(x)^(2*m),y(x), singsol=all)
\[ y \relax (x ) = \frac {b \sqrt {\cos ^{-2+2 m}\relax (x )}\, \left (\cos ^{-m +1}\relax (x )\right ) \cos \relax (x ) \left (\left (m -1\right ) \hypergeom \left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )-3 \hypergeom \left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin ^{2}\relax (x )\right )\right ) \left (-c_{1} \sin \left (b \sqrt {\cos ^{-2+2 m}\relax (x )}\, \left (\cos ^{-m +1}\relax (x )\right ) \sin \relax (x ) \hypergeom \left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin ^{2}\relax (x )\right )\right )+\cos \left (b \sqrt {\cos ^{-2+2 m}\relax (x )}\, \left (\cos ^{-m +1}\relax (x )\right ) \sin \relax (x ) \hypergeom \left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin ^{2}\relax (x )\right )\right )\right )}{3 c_{1} \cos \left (b \sqrt {\cos ^{-2+2 m}\relax (x )}\, \left (\cos ^{-m +1}\relax (x )\right ) \sin \relax (x ) \hypergeom \left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin ^{2}\relax (x )\right )\right )+3 \sin \left (b \sqrt {\cos ^{-2+2 m}\relax (x )}\, \left (\cos ^{-m +1}\relax (x )\right ) \sin \relax (x ) \hypergeom \left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin ^{2}\relax (x )\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.991 (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]
\begin{align*} 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) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(x)\right )}{m+1}+c_1\right ) \\ \end{align*}