55.2.27 problem 27

Internal problem ID [13253]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 27
Date solved : Wednesday, October 01, 2025 at 04:31:02 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 973

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 1.497 (sec). Leaf size: 1288

ode=D[y[x],x]==y[x]^2+(\[Alpha]*x+\[Beta])*y[x]+a*x^2+b*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") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*x**2 - b*x - c - (Alpha*x + BETA)*y(x) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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