Internal problem ID [9719]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\sinh \left (\lambda x \right ) y^{2} a -b \sinh \left (\lambda x \right ) \left (\cosh ^{n}\left (\lambda x \right )\right )=0} \end {gather*}
✓ 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)
\[ y \relax (x ) = \frac {\frac {\left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right ) \sqrt {b}\, c_{1} \BesselY \left (\frac {n +3}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right )}{\sqrt {a}\, \left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right )\right )}+\frac {\BesselJ \left (\frac {n +3}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right ) \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )-\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right ) c_{1} \lambda -\lambda \BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right )}{\left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b}\, \sqrt {a}\, \left (\cosh ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{\lambda \left (2+n \right )}\right )\right ) a}}{\cosh \left (\lambda x \right )} \]
✗ 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