55.2.8 problem 8

Internal problem ID [13234]
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 : 8
Date solved : Wednesday, October 01, 2025 at 03:44:37 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 170

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

\[ y = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {m +1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {m +1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )} \]

✓ Mathematica. Time used: 0.951 (sec). Leaf size: 1436

ode=D[y[x],x]==a*x^n*y[x]^2+b*x^m; 
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") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*x**n*y(x)**2 - b*x**m + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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