61.7.5 problem 5

Internal problem ID [12077]
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 : 5
Date solved : Wednesday, March 05, 2025 at 04:14:10 PM
CAS classification : [_Riccati]

\begin{align*} x y^{\prime }&=x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1} \end{align*}

✗ Maple

ode:=x*diff(y(x),x) = x*y(x)^2-a^2*x*ln(beta*x)^(2*k)+a*k*ln(beta*x)^(k-1); 
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*k)+a*k*(Log[\[Beta]*x])^(k-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
BETA = symbols("BETA") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(a**2*x*log(BETA*x)**(2*k) - a*k*log(BETA*x)**(k - 1) - 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*k) - a*k*log(BETA*x)**(k - 1)/x - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method