Internal problem ID [9805]
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: 50.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2} \cos \left (\lambda x \right ) a -b \cos \left (\lambda x \right ) \left (\sin ^{n}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 287
dsolve(diff(y(x),x)=a*cos(lambda*x)*y(x)^2+b*cos(lambda*x)*sin(lambda*x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (-\left (\sin ^{2+n}\left (\lambda x \right )\right ) c_{1} a b n -\left (\sin ^{2+n}\left (\lambda x \right )\right ) c_{1} a b \right ) \hypergeom \left (\left [\right ], \left [\frac {2 n +5}{2+n}\right ], -\frac {b a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )+\left (\lambda ^{2} n^{2} c_{1}+4 \lambda ^{2} n c_{1}+3 \lambda ^{2} c_{1}\right ) \hypergeom \left (\left [\right ], \left [\frac {n +3}{2+n}\right ], -\frac {b a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )\right ) \sin \left (\lambda x \right )+\left (-\left (\sin ^{2+n}\left (\lambda x \right )\right ) a b n -3 b a \left (\sin ^{2+n}\left (\lambda x \right )\right )\right ) \hypergeom \left (\left [\right ], \left [\frac {2 n +3}{2+n}\right ], -\frac {b a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )}{\left (n +1\right ) \lambda \sin \left (\lambda x \right ) \left (n +3\right ) a \left (c_{1} \sin \left (\lambda x \right ) \hypergeom \left (\left [\right ], \left [\frac {n +3}{2+n}\right ], -\frac {b a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )+\hypergeom \left (\left [\right ], \left [\frac {n +1}{2+n}\right ], -\frac {b a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==a*Cos[\[Lambda]*x]*y[x]^2+b*Cos[\[Lambda]*x]*Sin[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved