55.5.14 problem 14

Internal problem ID [13358]
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
Date solved : Wednesday, October 01, 2025 at 08:14:16 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 245

ode:=diff(y(x),x) = a*sinh(lambda*x)*y(x)^2+b*sinh(lambda*x)*cosh(lambda*x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-\lambda \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )+\cosh \left (\lambda x \right )^{\frac {n}{2}+1} \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) a \right ) \operatorname {sech}\left (\lambda x \right )}{a^{{3}/{2}} \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )} \]

✓ Mathematica. Time used: 0.457 (sec). Leaf size: 667

ode=D[y[x],x]==a*Sinh[\[Lambda]*x]*y[x]^2+b*Sinh[\[Lambda]*x]*Cosh[\[Lambda]*x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)**2*sinh(lambda_*x) - b*sinh(lambda_*x)*cosh(lambda_*x)**n + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(a*y(x)**2 + b*cosh(lambda_*x)**n)*sinh(lambda_*x) + Derivative(y(x), x) cannot be solved by the factorable group method