55.25.28 problem 28

Internal problem ID [13737]
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 : 28
Date solved : Thursday, October 02, 2025 at 06:39:48 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 50

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

\[ \left (A y+B x \right )^{-k} \left (a y-b x \right ) \left (A y-A b -B \left (a -x \right )\right )^{k -1}-c_1 = 0 \]

✓ Mathematica. Time used: 0.939 (sec). Leaf size: 62

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

\[ \text {Solve}\left [\frac {A b ((k-1) \log (-a B-A b+A y(x)+B x)+\log (b x-a y(x))-k \log (A y(x)+B x))}{a B+A b}=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out