Internal problem ID [10558]
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]
\[ \boxed {y^{\prime }-\cos \left (\lambda x \right ) y^{2} a=b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 284
dsolve(diff(y(x),x)=a*cos(lambda*x)*y(x)^2+b*cos(lambda*x)*sin(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (\sin \left (\lambda x \right )^{n +2} c_{1} a b n +\sin \left (\lambda x \right )^{n +2} c_{1} a b \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2 n +5}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a b}{\lambda ^{2} \left (n +2\right )^{2}}\right )+\left (-c_{1} \lambda ^{2} n^{2}-4 c_{1} \lambda ^{2} n -3 c_{1} \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a b}{\lambda ^{2} \left (n +2\right )^{2}}\right )\right ) \sin \left (\lambda x \right )+\left (\sin \left (\lambda x \right )^{n +2} a b n +3 \sin \left (\lambda x \right )^{n +2} a b \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3+2 n}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a b}{\lambda ^{2} \left (n +2\right )^{2}}\right )}{\left (1+n \right ) \lambda \sin \left (\lambda x \right ) \left (n +3\right ) a \left (c_{1} \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a b}{\lambda ^{2} \left (n +2\right )^{2}}\right )+\operatorname {hypergeom}\left (\left [\right ], \left [\frac {1+n}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a b}{\lambda ^{2} \left (n +2\right )^{2}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.376 (sec). Leaf size: 633
DSolve[y'[x]==a*Cos[\[Lambda]*x]*y[x]^2+b*Cos[\[Lambda]*x]*Sin[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\csc (\lambda x) \left (-\lambda \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(\lambda x) \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \left (\operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )-c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{2 a \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )} y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}-\lambda \csc (\lambda x)}{2 a} \end{align*}