55.5.9 problem 9

Internal problem ID [13353]
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 : 9
Date solved : Wednesday, October 01, 2025 at 08:07:20 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 73

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

\[ y = \frac {b \int {\mathrm e}^{\frac {-2 b x \beta +a \sinh \left (\beta x \right )}{\beta }}d x -b c_1 +{\mathrm e}^{\frac {-2 b x \beta +a \sinh \left (\beta x \right )}{\beta }}}{-\int {\mathrm e}^{\frac {-2 b x \beta +a \sinh \left (\beta x \right )}{\beta }}d x +c_1} \]

✓ Mathematica. Time used: 2.703 (sec). Leaf size: 298

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

\begin{align*} y(x)&\to -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )+b \int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+b c_1}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+c_1}\\ y(x)&\to -b\\ y(x)&\to -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]}-b \end{align*}

✗ Sympy

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

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