61.8.3 problem 12

Internal problem ID [12084]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 12
Date solved : Wednesday, March 05, 2025 at 04:14:43 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

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