55.19.28 problem 28

Internal problem ID [13513]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 28
Date solved : Wednesday, October 01, 2025 at 03:49:27 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 227

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

\[ y = \frac {-x \left (\ln \left (x \right )-1\right ) {\mathrm e}^{\int \frac {\ln \left (x \right )^{2} f \left (x \right ) a \,x^{2}+\left (-2 f \left (x \right ) a \,x^{2}-2\right ) \ln \left (x \right )+f \left (x \right ) a \,x^{2}}{x \left (\ln \left (x \right )-1\right )}d x}+c_1 a -\int \ln \left (x \right ) {\mathrm e}^{a \int \frac {x \ln \left (x \right )^{2} f \left (x \right )}{\ln \left (x \right )-1}d x -2 a \int \frac {x \ln \left (x \right ) f \left (x \right )}{\ln \left (x \right )-1}d x +a \int \frac {x f \left (x \right )}{\ln \left (x \right )-1}d x -2 \int \frac {\ln \left (x \right )}{x \left (\ln \left (x \right )-1\right )}d x}d x}{a x \left (\ln \left (x \right )-1\right ) \left (c_1 a -\int \ln \left (x \right ) {\mathrm e}^{a \int \frac {x \ln \left (x \right )^{2} f \left (x \right )}{\ln \left (x \right )-1}d x -2 a \int \frac {x \ln \left (x \right ) f \left (x \right )}{\ln \left (x \right )-1}d x +a \int \frac {x f \left (x \right )}{\ln \left (x \right )-1}d x -2 \int \frac {\ln \left (x \right )}{x \left (\ln \left (x \right )-1\right )}d x}d x \right )} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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