2.24.22 Problem 36
Internal
problem
ID
[13586]
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
:
36
Date
solved
:
Friday, December 19, 2025 at 07:38:30 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}}&=\frac {a^{2} \left (x -1\right ) \left (3 x -1\right )}{x^{7}} \\
\end{align*}
Unknown ode type.
2.24.22.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 167
ode:=y(x)*diff(y(x),x)-a*(5*x-4)/x^4*y(x) = a^2*(x-1)*(3*x-1)/x^7;
dsolve(ode,y(x), singsol=all);
\[
c_1 -\frac {9 \,2^{{2}/{3}} \sqrt {\frac {x^{3} y+a x -a}{\left (x^{2} y+a \right ) x}}\, 5^{{1}/{6}} \left (x -\frac {3}{4}\right )}{5 x {\left (-\frac {a}{x \left (x^{2} y+a \right )}\right )}^{{1}/{3}} \left (\frac {3 x^{3} y+3 a x -a}{\left (x^{2} y+a \right ) x}\right )^{{1}/{6}}}-729 \int _{}^{\frac {\frac {9 x^{3} y}{5}+\frac {9 a x}{5}-\frac {27 a}{20}}{\left (x^{2} y+a \right ) x}}\frac {\textit {\_a} \sqrt {20 \textit {\_a} -9}}{\left (5 \textit {\_a} -9\right )^{{1}/{3}} \left (4 \textit {\_a} +9\right )^{{1}/{6}} \left (400 \textit {\_a}^{3}-1701 \textit {\_a} +729\right )}d \textit {\_a} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {a \left (5 x -4\right ) y \left (x \right )}{x^{4}}=\frac {a^{2} \left (x -1\right ) \left (3 x -1\right )}{x^{7}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {a \left (5 x -4\right ) y \left (x \right )}{x^{4}}+\frac {a^{2} \left (x -1\right ) \left (3 x -1\right )}{x^{7}}}{y \left (x \right )} \end {array} \]
2.24.22.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-a*(5*x-4)*x^(-4)*y[x]==a^2*(x-1)*(3*x-1)*x^(-7);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.22.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(x - 1)*(3*x - 1)/x**7 - a*(5*x - 4)*y(x)/x**4 + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out