61.7.4 problem 4

Internal problem ID [12076]
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 : 4
Date solved : Sunday, March 30, 2025 at 10:38:47 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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