61.4.11 problem 32

Internal problem ID [12037]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and exponential functions
Problem number : 32
Date solved : Wednesday, March 05, 2025 at 03:54:35 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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