2.24.8 Problem 8
Internal
problem
ID
[13572]
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
:
8
Date
solved
:
Friday, December 19, 2025 at 07:09:23 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=x^{n -1} \left (\left (1+2 n \right ) x +a n \right ) y-n \,x^{2 n} \left (a +x \right ) \\
\end{align*}
Unknown ode type.
2.24.8.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 153
ode:=y(x)*diff(y(x),x) = x^(n-1)*((1+2*n)*x+a*n)*y(x)-n*x^(2*n)*(x+a);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {2 \left (-\frac {\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x +2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )}{2}+\left (a +\frac {x}{2}\right ) n \right ) x^{n}}{\tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x +2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}-n}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=x^{13571} \left (27145 x +13572 a \right ) y \left (x \right )-13572 x^{27144} \left (x +a \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x^{13571} \left (27145 x +13572 a \right ) y \left (x \right )-13572 x^{27144} \left (x +a \right )}{y \left (x \right )} \end {array} \]
2.24.8.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==x^(n-1)*((1+2*n)*x+a*n)*y[x]-n*x^(2*n)*(x+a);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.8.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(n*x**(2*n)*(a + x) - 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