55.26.12 problem 12

Internal problem ID [13759]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 12
Date solved : Thursday, October 02, 2025 at 07:58:28 AM
CAS classification : [_Abel]

\begin{align*} 9 y^{\prime }&=-x^{m} \left (a \,x^{1-m}+b \right )^{2 \lambda +1} y^{3}-x^{-2 m} \left (9 a +2+9 b m \,x^{m -1}\right ) \left (a \,x^{1-m}+b \right )^{-\lambda -2} \end{align*}

✗ Maple

ode:=9*diff(y(x),x) = -x^m*(a*x^(-m+1)+b)^(2*lambda+1)*y(x)^3-x^(-2*m)*(9*a+2+9*b*m*x^(m-1))*(a*x^(-m+1)+b)^(-lambda-2); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=9*D[y[x],x]==-x^m*(a*x^(1-m)+b)^(2*\[Lambda]+1)*y[x]^3-x^(-2*m)*(9*a+2+9*b*m*x^(m-1))*(a*x^(1-m)+b)^(-\[Lambda]-2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
y = Function("y") 
ode = Eq(x**m*(a*x**(1 - m) + b)**(2*lambda_ + 1)*y(x)**3 + 9*Derivative(y(x), x) + (a*x**(1 - m) + b)**(-lambda_ - 2)*(9*a + 9*b*m*x**(m - 1) + 2)/x**(2*m),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out