61.4.3 problem 24

Internal problem ID [12029]
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 : 24
Date solved : Sunday, March 30, 2025 at 10:19:29 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n} \end{align*}

✗ Maple

ode:=diff(y(x),x) = a*exp(lambda*x)*y(x)^2+b*n*x^(n-1)-a*b^2*exp(lambda*x)*x^(2*n); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]^2+b*n*x^(n-1)-a*b^2*Exp[\[Lambda]*x]*x^(2*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") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*b**2*x**(2*n)*exp(lambda_*x) - a*y(x)**2*exp(lambda_*x) - b*n*x**(n - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*b**2*x**(2*n)*exp(lambda_*x) - a*y(x)**2*exp(lambda_*x) - b*n*x**(n - 1) + Derivative(y(x), x) cannot be solved by the lie group method