61.7.6 problem 6

Internal problem ID [12078]
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 : 6
Date solved : Wednesday, March 05, 2025 at 04:14:17 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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