61.5.13 problem 13

Internal problem ID [12058]
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 : 13
Date solved : Sunday, March 30, 2025 at 10:25:46 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4} \end{align*}

✗ Maple

ode:=diff(y(x),x) = y(x)^2-lambda^2+a*cosh(lambda*x)^n*sinh(lambda*x)^(-n-4); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2-\[Lambda]^2+a*Cosh[\[Lambda]*x]^n*Sinh[\[Lambda]*x]^(-n-4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*sinh(lambda_*x)**(-n - 4)*cosh(lambda_*x)**n + lambda_**2 - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*sinh(lambda_*x)**(-n - 4)*cosh(lambda_*x)**n + lambda_**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method