61.3.15 problem 15

Internal problem ID [12020]
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 : 15
Date solved : Sunday, March 30, 2025 at 10:17:04 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 485

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

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

✓ Mathematica. Time used: 5.191 (sec). Leaf size: 902

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

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

✗ Sympy

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

NotImplementedError : The given ODE -a*b*y(x)*exp(x*(lambda_ + mu)) - a*y(x)**2*exp(mu*x) + b*lambda_*exp(lambda_*x) + Derivative(y(x), x) cannot be solved by the lie group method