2.24.40 Problem 59
Internal
problem
ID
[13604]
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
1.3.3-2.
Problem
number
:
59
Date
solved
:
Friday, December 19, 2025 at 08:15:27 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-\left (\left (2 n -1\right ) x -a n \right ) x^{-n -1} y&=n \left (-a +x \right ) x^{-2 n} \\
\end{align*}
Unknown ode type.
2.24.40.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 151
ode:=y(x)*diff(y(x),x)-((2*n-1)*x-a*n)*x^(-1-n)*y(x) = n*(x-a)*x^(-2*n);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 x^{-n} \left (-\frac {\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )}{2}+\left (a -\frac {x}{2}\right ) n \right )}{\tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}+n}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {\left (27207 x -13604 a \right ) y \left (x \right )}{x^{13605}}=\frac {13604 \left (x -a \right )}{x^{27208}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {\left (27207 x -13604 a \right ) y \left (x \right )}{x^{13605}}+\frac {13604 \left (x -a \right )}{x^{27208}}}{y \left (x \right )} \end {array} \]
2.24.40.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-((2*n-1)*x-a*n)*x^(-n-1)*y[x]==n*(x-a)*x^(-2*n);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.40.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(-n*(-a + x)/x**(2*n) - x**(-n - 1)*(-a*n + x*(2*n - 1))*y(x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out