55.5.4 problem 4

Internal problem ID [13348]
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 : 4
Date solved : Wednesday, October 01, 2025 at 07:56:50 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 51

ode:=diff(y(x),x) = lambda*sinh(lambda*x)*y(x)^2-lambda*sinh(lambda*x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {2 \left ({\mathrm e}^{\frac {\cosh \left (2 \lambda x \right )}{2}+\frac {1}{2}} c_1 -\frac {\cosh \left (\lambda x \right ) \sqrt {\pi }\, \left (\operatorname {erfi}\left (\cosh \left (\lambda x \right )\right ) c_1 +1\right )}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erfi}\left (\cosh \left (\lambda x \right )\right ) c_1 +1\right )} \]

✗ Mathematica

ode=D[y[x],x]==\[Lambda]*Sinh[\[Lambda]*x]*y[x]^2-\[Lambda]*Sinh[\[Lambda]*x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-lambda_*y(x)**2*sinh(lambda_*x) + lambda_*sinh(lambda_*x)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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