2.25.12 Problem 20

Internal problem ID [13627]
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 : 20
Date solved : Friday, December 19, 2025 at 09:04:21 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.25.12.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 258

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

\[ -\left (B -c \right )^{2} \left (A \beta +k \right ) \int _{}^{\frac {\left (B -c \right ) x}{-y A +A \beta -B x}}\textit {\_a}^{\frac {-A d -2 B^{2}+3 B c -c^{2}}{\left (B -c \right )^{2}}} {\mathrm e}^{-\frac {1}{\textit {\_a}}} {\left (\left (A d +B^{2}-B c \right ) \textit {\_a} +\left (B -c \right )^{2}\right )}^{\frac {A d +B c -c^{2}}{\left (B -c \right )^{2}}}d \textit {\_a} +{\mathrm e}^{\frac {y A -A \beta +B x}{\left (B -c \right ) x}} \left (\frac {\left (B -c \right ) x}{-y A +A \beta -B x}\right )^{\frac {-A d -B^{2}+B c}{\left (B -c \right )^{2}}} x \left (B -c \right ) \left (\frac {\left (B -c \right ) \left (\left (-B +c \right ) y+B \beta -c \beta +d x \right ) A}{-y A +A \beta -B x}\right )^{\frac {A d +B^{2}-B c}{\left (B -c \right )^{2}}}+c_1 = 0 \]

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

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

2.25.12.2 ✗ Mathematica

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

Not solved

2.25.12.3 ✗ Sympy

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

Timed Out