2.24.9 Problem 9
Internal
problem
ID
[13573]
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
:
9
Date
solved
:
Friday, December 19, 2025 at 07:11:56 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=a \left (-b n +x \right ) x^{n -1} y+c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1} \\
\end{align*}
Unknown ode type.
2.24.9.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 1972
ode:=y(x)*diff(y(x),x) = a*(-b*n+x)*x^(n-1)*y(x)+c*(x^2-(1+2*n)*b*x+n*(n+1)*b^2)*x^(2*n-1);
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=a \left (-13573 b +x \right ) x^{13572} y \left (x \right )+c \left (184239902 b^{2}-27147 b x +x^{2}\right ) x^{27145} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \left (-13573 b +x \right ) x^{13572} y \left (x \right )+c \left (184239902 b^{2}-27147 b x +x^{2}\right ) x^{27145}}{y \left (x \right )} \end {array} \]
2.24.9.2 ✓ Mathematica. Time used: 0.31 (sec). Leaf size: 200
ode=y[x]*D[y[x],x]==a*(x-n*b)*x^(n-1)*y[x]+c*(x^2-(2*n+1)*b*x+n*(n+1)*b^2)*x^(2*n-1);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {a^2 \left (-\frac {2 a \text {arctanh}\left (\frac {a^2-\frac {2 a c (n+1) y(x)}{-b c x^n-b c n x^n+c x^{n+1}}}{a \sqrt {a^2+4 c (n+1)}}\right )}{\sqrt {a^2+4 c (n+1)}}-\log \left (a^2 \left (\frac {a y(x)}{-b c x^n-b c n x^n+c x^{n+1}}+1\right )-\frac {a^2 c (n+1) y(x)^2}{\left (-b c x^n-b c n x^n+c x^{n+1}\right )^2}\right )\right )}{2 c (n+1)}=\frac {a^2 (\log (x-b (n+1))+n \log (x))}{c (n+1)}+c_1,y(x)\right ]
\]
2.24.9.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(-a*x**(n - 1)*(-b*n + x)*y(x) - c*x**(2*n - 1)*(b**2*n*(n + 1) - b*x*(2*n + 1) + x**2) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out