2.25.7 Problem 9
Internal
problem
ID
[13622]
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
:
9
Date
solved
:
Friday, December 19, 2025 at 08:57:14 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
\left (y a x -a k y+b x -b k \right ) y^{\prime }&=y^{2} c +d x y+\left (-d k +b \right ) y \\
\end{align*}
Unknown ode type.
2.25.7.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 131
ode:=(a*x*y(x)-a*k*y(x)+b*x-b*k)*diff(y(x),x) = c*y(x)^2+d*x*y(x)+(-d*k+b)*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {2 \left (k -x \right ) d}{\tan \left (\frac {\operatorname {RootOf}\left (-2 a d k \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 a d x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}\, a b +a b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-b c \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 a b \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}}{2}\right ) a \sqrt {-\frac {\left (a -c \right )^{2}}{a^{2}}}+a -c}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a x y \left (x \right )-a k y \left (x \right )+b x -b k \right ) \left (\frac {d}{d x}y \left (x \right )\right )=c y \left (x \right )^{2}+d x y \left (x \right )+\left (-d k +b \right ) y \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {c y \left (x \right )^{2}+d x y \left (x \right )+\left (-d k +b \right ) y \left (x \right )}{a x y \left (x \right )-a k y \left (x \right )+b x -b k} \end {array} \]
2.25.7.2 ✗ Mathematica
ode=(a*x*y[x]-a*k*y[x]+b*x-b*k)*D[y[x],x]==c*y[x]^2+d*x*y[x]+(b-d*k)*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.25.7.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
k = symbols("k")
c = symbols("c")
d = symbols("d")
y = Function("y")
ode = Eq(-c*y(x)**2 - d*x*y(x) - (b - d*k)*y(x) + (-a*k*y(x) + a*x*y(x) - b*k + b*x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out