2.25.17 Problem 34
Internal
problem
ID
[13632]
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
:
34
Date
solved
:
Friday, December 19, 2025 at 09:20:10 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
x \left (\left (m -1\right ) \left (A x +B \right ) y+m \left (d \,x^{2}+e x +F \right )\right ) y^{\prime }&=\left (A \left (1-n \right ) x -B n \right ) y^{2}+\left (d \left (2-n \right ) x^{2}+e \left (1-n \right ) x -F n \right ) y \\
\end{align*}
Unknown ode type.
2.25.17.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 2371
ode:=x*((m-1)*(A*x+B)*y(x)+m*(d*x^2+e*x+F))*diff(y(x),x) = (A*(1-n)*x-B*n)*y(x)^2+(d*(2-n)*x^2+e*(1-n)*x-F*n)*y(x);
dsolve(ode,y(x), singsol=all);
\[
\text {Expression too large to display}
\]
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
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\left (m -1\right ) \left (A x +B \right ) y \left (x \right )+m \left (d \,x^{2}+e x +F \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (-13631 A x -13632 B \right ) y \left (x \right )^{2}+\left (-13630 d \,x^{2}-13631 e x -13632 F \right ) y \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (-13631 A x -13632 B \right ) y \left (x \right )^{2}+\left (-13630 d \,x^{2}-13631 e x -13632 F \right ) y \left (x \right )}{x \left (\left (m -1\right ) \left (A x +B \right ) y \left (x \right )+m \left (d \,x^{2}+e x +F \right )\right )} \end {array} \]
2.25.17.2 ✓ Mathematica. Time used: 143.711 (sec). Leaf size: 41
ode=x*( (m-1)*(A*x+B)*y[x]+m*(d*x^2+e*x+F) )*D[y[x],x]==( A*(1-n)*x-B*n)*y[x]^2+ (d*(2-n)*x^2+e*(1-n)*x-F*n)*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-A \log \left (A x y(x)+B y(x)+d x^2+e x+F\right )+A m \log (y(x))+A n \log (x)=c_1,y(x)\right ]
\]
2.25.17.3 ✗ Sympy
from sympy import *
x = symbols("x")
m = symbols("m")
A = symbols("A")
B = symbols("B")
d = symbols("d")
e = symbols("e")
F = symbols("F")
n = symbols("n")
y = Function("y")
ode = Eq(x*(m*(F + d*x**2 + e*x) + (m - 1)*(A*x + B)*y(x))*Derivative(y(x), x) - (A*x*(1 - n) - B*n)*y(x)**2 - (-F*n + d*x**2*(2 - n) + e*x*(1 - n))*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out