55.5.8 problem 8

Internal problem ID [13352]
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 : 8
Date solved : Wednesday, October 01, 2025 at 08:06:22 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

ode:=diff(y(x),x) = alpha*y(x)^2+beta+gamma*cosh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {i \left (c_1 \operatorname {MathieuSPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+\operatorname {MathieuCPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )}{2 \alpha \left (c_1 \operatorname {MathieuS}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+\operatorname {MathieuC}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )} \]

✓ Mathematica. Time used: 0.162 (sec). Leaf size: 140

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

\begin{align*} y(x)&\to -\frac {i c_1 \text {MathieuCPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]-i \text {MathieuSPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]}{2 \alpha c_1 \text {MathieuC}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]-2 \alpha \text {MathieuS}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]}\\ y(x)&\to -\frac {i \text {MathieuCPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]}{2 \alpha \text {MathieuC}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]} \end{align*}

✗ Sympy

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

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