2.23.6 Problem 6
Internal
problem
ID
[13559]
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
1.3.2.
Problem
number
:
6
Date
solved
:
Friday, December 19, 2025 at 06:55:46 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }&=\left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y+1 \\
\end{align*}
Unknown ode type.
2.23.6.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 187
ode:=y(x)*diff(y(x),x) = (a/x^(2/3)-2/3/a/x^(1/3))*y(x)+1;
dsolve(ode,y(x), singsol=all);
\[
\frac {-\operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, c_1 \,a^{2}+\operatorname {BesselK}\left (1, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) c_1 -x^{{1}/{3}} \operatorname {BesselK}\left (0, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )}{-\operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y \left (x \right )+1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y \left (x \right )+1}{y \left (x \right )} \end {array} \]
2.23.6.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y[x]+1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.23.6.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq((-a/x**(2/3) + 2/(3*a*x**(1/3)))*y(x) + y(x)*Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0