Internal problem ID [9715]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\sinh \left (\lambda x \right ) y^{2} a -b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 333
dsolve(diff(y(x),x)=a*sinh(lambda*x)*y(x)^2+b*sinh(lambda*x)*cosh(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\frac {\sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1} c_{1} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )}{\sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )}+\frac {\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} \lambda -\lambda \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) a}}{\cosh \left (\lambda x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==a*Sinh[\[Lambda]*x]*y[x]^2+b*Sinh[\[Lambda]*x]*Cosh[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved