55.25.31 problem 31

Internal problem ID [13740]
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 : 31
Date solved : Thursday, October 02, 2025 at 06:56:01 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 154

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

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

✓ Mathematica. Time used: 29.074 (sec). Leaf size: 173

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

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

✗ Sympy

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

Timed Out