61.3.10 problem 10

Internal problem ID [12015]
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 : 10
Date solved : Wednesday, March 05, 2025 at 03:50:52 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 4.344 (sec). Leaf size: 837

ode=D[y[x],x]==b*Exp[\[Mu]*x]*y[x]^2+a*\[Lambda]*Exp[\[Lambda]*x]-a^2*b*Exp[(\[Mu]+2*\[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") 
cg = symbols("cg") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(a**2*b*exp(x*(2*cg + mu)) - a*cg*exp(cg*x) - b*y(x)**2*exp(mu*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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