61.5.1 problem 1

Internal problem ID [12046]
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 : 1
Date solved : Sunday, March 30, 2025 at 10:21:15 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 409

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

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

✓ Mathematica. Time used: 6.242 (sec). Leaf size: 212

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

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

✗ Sympy

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

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