55.2.9 problem 9

Internal problem ID [13235]
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 : 9
Date solved : Wednesday, October 01, 2025 at 03:45:37 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+k \left (a x +b \right )^{n} \left (c x +d \right )^{-n -4} \end{align*}

✗ Maple

ode:=diff(y(x),x) = y(x)^2+k*(a*x+b)^n*(c*x+d)^(-n-4); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+k*(a*x+b)^n*(c*x+d)^(-n-4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
k = symbols("k") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-k*(a*x + b)**n*(c*x + d)**(-n - 4) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -k*(a*x + b)**n*(c*x + d)**(-n - 4) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method