2.25.13 Problem 21

Internal problem ID [13628]
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 : 21
Date solved : Friday, December 19, 2025 at 09:07:46 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.25.13.1 ✓ Maple. Time used: 0.005 (sec). Leaf size: 770

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

\begin{align*} \text {Solution too large to show}\end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
Looking for potential symmetries 
Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (A x y \left (x \right )+A k y \left (x \right )+B \,x^{2}+B k x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=c y \left (x \right )^{2}+d x y \left (x \right )+k \left (d -B \right ) y \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {c y \left (x \right )^{2}+d x y \left (x \right )+k \left (d -B \right ) y \left (x \right )}{A x y \left (x \right )+A k y \left (x \right )+B \,x^{2}+B k x} \end {array} \]

2.25.13.2 ✗ Mathematica

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

Not solved

2.25.13.3 ✗ Sympy

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

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0