55.3.19 problem 19

Internal problem ID [13323]
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 : 19
Date solved : Wednesday, October 01, 2025 at 07:19:13 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

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

\[ y = b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{-\mu x} \mu \left (-1+\frac {1}{c_1 \mu \,{\mathrm e}^{\mu x}+1}\right ) \]

✓ Mathematica. Time used: 0.597 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to b e^{\lambda x}+\frac {\mu }{-e^{\mu x}+c_1 \mu }\\ y(x)&\to b e^{\lambda x} \end{align*}

✗ Sympy

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

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