55.7.7 problem 7

Internal problem ID [13378]
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 : 7
Date solved : Wednesday, October 01, 2025 at 09:16:48 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 454

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.388 (sec). Leaf size: 868

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

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