Internal problem ID [10473]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-1. Equations with hyperbolic
sine and cosine
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} \cosh \left (\lambda x \right ) a=b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 333
dsolve(diff(y(x),x)=a*cosh(lambda*x)*y(x)^2+b*cosh(lambda*x)*sinh(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\frac {\sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1} c_{1} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )}{\sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )}+\frac {\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} \lambda -\lambda \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \sinh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) a}}{\sinh \left (\lambda x \right )} \]
✓ Solution by Mathematica
Time used: 1.277 (sec). Leaf size: 633
DSolve[y'[x]==a*Cosh[\[Lambda]*x]*y[x]^2+b*Cosh[\[Lambda]*x]*Sinh[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\text {csch}(\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} \sinh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} \sinh ^{\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} \sinh ^{\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} \sinh ^{\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} \sinh ^{\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} \sinh ^{\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} \sinh ^{\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} \sinh ^{\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} \sinh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )} y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \sinh ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sinh ^{\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} \sinh ^{\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} \sinh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}-\lambda \text {csch}(\lambda x)}{2 a} \end{align*}