55.7.1 problem 1

Internal problem ID [13372]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-1. Equations Containing Logarithmic Functions
Problem number : 1
Date solved : Wednesday, October 01, 2025 at 08:58:25 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==a*(Log[x])^n*y[x]^2+b*m*x^(m-1)-a*b^2*x^(2*m)*(Log[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") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*b**2*x**(2*m)*log(x)**n - a*y(x)**2*log(x)**n - b*m*x**(m - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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