55.2.43 problem 43

Internal problem ID [13269]
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 : 43
Date solved : Wednesday, October 01, 2025 at 04:57:35 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 72

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

\[ y = \frac {\left (\tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (b \,x^{n}+c_1 n \right )}{2 b^{2} n}\right ) \sqrt {4 b^{2} a c -b^{4}}-b^{2}\right ) x^{-n}}{2 a b} \]

✓ Mathematica. Time used: 0.477 (sec). Leaf size: 118

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

\begin{align*} y(x)&\to \frac {x^{-n} \left (-b+\frac {\sqrt {b^2-4 a c} \left (-e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1\right )}{e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1}\right )}{2 a}\\ y(x)&\to \frac {x^{-n} \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \end{align*}

✗ Sympy

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

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