55.25.37 problem 37

Internal problem ID [13746]
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 : 37
Date solved : Thursday, October 02, 2025 at 07:55:10 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 236

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

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

✓ Mathematica. Time used: 47.235 (sec). Leaf size: 226

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

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

✗ Sympy

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

Timed Out