Internal problem ID [10556]
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: 48.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} \sin \left (\lambda x \right ) a=b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 334
dsolve(diff(y(x),x)=a*sin(lambda*x)*y(x)^2+b*sin(lambda*x)*cos(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\frac {\sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1} c_{1} \lambda \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a}-\frac {\left (\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) \lambda }{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a}}{\cos \left (\lambda x \right )} \]
✓ Solution by Mathematica
Time used: 1.409 (sec). Leaf size: 695
DSolve[y'[x]==a*Sin[\[Lambda]*x]*y[x]^2+b*Sin[\[Lambda]*x]*Cos[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\lambda \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}{2 a \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+2 a c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )} y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\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} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}+\lambda \sec (\lambda x)}{2 a} \end{align*}