61.3.6 problem 6

Internal problem ID [12011]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential Functions
Problem number : 6
Date solved : Sunday, March 30, 2025 at 10:15:24 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 73

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

\[ y = \frac {b c_1 -b \int {\mathrm e}^{\frac {2 b x \lambda +{\mathrm e}^{\lambda x} a}{\lambda }}d x +{\mathrm e}^{\frac {2 b x \lambda +{\mathrm e}^{\lambda x} a}{\lambda }}}{-\int {\mathrm e}^{\frac {2 b x \lambda +{\mathrm e}^{\lambda x} a}{\lambda }}d x +c_1} \]

✓ Mathematica. Time used: 0.503 (sec). Leaf size: 115

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

\begin{align*} y(x)\to \frac {b \left (-2 \lambda e^{\frac {a e^{\lambda x}}{\lambda }} \left (-\frac {a e^{\lambda x}}{\lambda }\right )^{\frac {2 b}{\lambda }}+2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }\right )}{2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }} \\ y(x)\to b \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a*b*exp(lambda_*x) - a*y(x)*exp(lambda_*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*exp(lambda_*x) - a*y(x)*exp(lambda_*x) + b**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method