55.26.10 problem 10

Internal problem ID [13757]
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 : 10
Date solved : Thursday, October 02, 2025 at 07:58:14 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=a \,x^{n} y^{3}+3 a b \,x^{n +m} y^{2}-b m \,x^{m -1}-2 a \,b^{3} x^{n +3 m} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 9748

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

\begin{align*} y &= -x^{m} b \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

✓ Mathematica. Time used: 15.134 (sec). Leaf size: 306

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

\begin{align*} y(x)&\to -b x^m-\frac {e^{-\frac {3 a b^2 x^{2 m+n+1}}{2 m+n+1}}}{\sqrt {\frac {3^{-\frac {n+1}{2 m+n+1}} 4^{\frac {m}{2 m+n+1}} x^{-2 m} \left (\frac {a b^2 x^{2 m+n+1}}{2 m+n+1}\right )^{\frac {2 m}{2 m+n+1}} \Gamma \left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )}{b^2}+c_1}}\\ y(x)&\to -b x^m+\frac {e^{-\frac {3 a b^2 x^{2 m+n+1}}{2 m+n+1}}}{\sqrt {\frac {3^{-\frac {n+1}{2 m+n+1}} 4^{\frac {m}{2 m+n+1}} x^{-2 m} \left (\frac {a b^2 x^{2 m+n+1}}{2 m+n+1}\right )^{\frac {2 m}{2 m+n+1}} \Gamma \left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )}{b^2}+c_1}}\\ y(x)&\to -b x^m \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE 2*a*b**3*x**(3*m + n) - 3*a*b*x**(m + n)*y(x)**2 - a*x**n*y(x)**3 + b*m*x**(m - 1) + Derivative(y(x), x) cannot be solved by the lie group method