55.25.14 problem 14

Internal problem ID [13723]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.4-2.
Problem number : 14
Date solved : Thursday, October 02, 2025 at 04:32:58 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 134

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

\begin{align*} y &= \frac {-b x m +x^{2-m} \sqrt {2}\, \sqrt {\left (A \,x^{2 m} a +\left (\frac {b^{2} m}{2}+2 a c \right ) x^{m}-2 c_1 a m \right ) m \,x^{m -2}}}{2 a m} \\ y &= \frac {-b x m -x^{2-m} \sqrt {2}\, \sqrt {\left (A \,x^{2 m} a +\left (\frac {b^{2} m}{2}+2 a c \right ) x^{m}-2 c_1 a m \right ) m \,x^{m -2}}}{2 a m} \\ \end{align*}

✓ Mathematica. Time used: 33.865 (sec). Leaf size: 161

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

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

✗ Sympy

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

Timed Out