2.22.3 Problem 3
Internal
problem
ID
[13498]
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
1.3.1-2.
Solvable
equations
and
their
solutions
Problem
number
:
3
Date
solved
:
Friday, December 19, 2025 at 05:03:49 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \\
\end{align*}
Unknown ode type.
2.22.3.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 90
ode:=y(x)*diff(y(x),x)-y(x) = -2/9*x+A+B/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {9 \left (A \sqrt {x}+B -\frac {2 x^{{3}/{2}}}{9}\right ) A}{3 A \sqrt {x}+3 \operatorname {RootOf}\left (18 A^{3} \int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_a}^{3} B^{2}+9 \textit {\_a} \,A^{3}-9 A^{3}}d \textit {\_a} -9 A \int \frac {1}{9 A x -2 x^{2}+9 B \sqrt {x}}d x +2 c_1 \right ) B}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}}}{y \left (x \right )} \end {array} \]
2.22.3.2 ✓ Mathematica. Time used: 4.759 (sec). Leaf size: 415
ode=y[x]*D[y[x],x]-y[x]==-2/9*x+A+B*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [6 \text {RootSum}\left [8 \text {$\#$1}^6-72 \text {$\#$1}^4 A-36 \text {$\#$1}^4 y(x)-72 \text {$\#$1}^3 B+162 \text {$\#$1}^2 A^2+162 \text {$\#$1}^2 A y(x)+54 \text {$\#$1}^2 y(x)^2+324 \text {$\#$1} A B+162 \text {$\#$1} B y(x)-81 A y(x)^2+162 B^2-27 y(x)^3\&,\frac {-2 \text {$\#$1}^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} A \log \left (\sqrt {x}-\text {$\#$1}\right )+9 B \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} y(x) \log \left (\sqrt {x}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-48 \text {$\#$1}^3 A-24 \text {$\#$1}^3 y(x)-36 \text {$\#$1}^2 B+54 \text {$\#$1} A^2+54 \text {$\#$1} A y(x)+18 \text {$\#$1} y(x)^2+54 A B+27 B y(x)}\&\right ]+\int _1^{y(x)}\left (\frac {162 K[1]}{8 x^3-72 A x^2-36 K[1] x^2-72 B x^{3/2}+162 A^2 x+54 K[1]^2 x+162 A K[1] x+324 A B \sqrt {x}+162 B K[1] \sqrt {x}-27 K[1]^3+162 B^2-81 A K[1]^2}+\frac {162 K[1]}{-8 x^3+72 A x^2+36 K[1] x^2+72 B x^{3/2}-162 A^2 x-54 K[1]^2 x-162 A K[1] x-324 A B \sqrt {x}-162 B K[1] \sqrt {x}+27 K[1]^3-162 B^2+81 A K[1]^2}\right )dK[1]=c_1,y(x)\right ]
\]
2.22.3.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
B = symbols("B")
y = Function("y")
ode = Eq(-A - B/sqrt(x) + 2*x/9 + y(x)*Derivative(y(x), x) - y(x),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