2.22.11 Problem 14
Internal
problem
ID
[13506]
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
:
14
Date
solved
:
Friday, December 19, 2025 at 05:13:02 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-y&=\frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \\
\end{align*}
Unknown ode type.
2.22.11.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 177
ode:=y(x)*diff(y(x),x)-y(x) = 4/9*x+2*A*x^2+2*A^2*x^3;
dsolve(ode,y(x), singsol=all);
\[
-\frac {27 \left (3^{{1}/{4}} A y \left (A x +\frac {1}{3}\right ) {\left (\frac {A \left (3 A \,x^{2}+3 y+x \right ) \left (9 A^{2} x^{2}-9 A y+9 A x +2\right )}{\left (-9 A y+3 A x +1\right )^{2}}\right )}^{{1}/{4}}-\frac {\left (\frac {1}{3}-3 A y+A x \right ) \left (\int _{}^{\frac {\left (3 A x +1\right )^{2}}{-9 A y+3 A x +1}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_1 \right ) \sqrt {\frac {\left (3 A x +1\right )^{2}}{-9 A y+3 A x +1}}}{9}\right )}{\sqrt {\frac {\left (3 A x +1\right )^{2}}{-9 A y+3 A x +1}}\, \left (-9 A y+3 A x +1\right )} = 0
\]
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
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {4 x}{9}+2 A \,x^{2}+2 A^{2} x^{3}}{y \left (x \right )} \end {array} \]
2.22.11.2 ✓ Mathematica. Time used: 1.171 (sec). Leaf size: 170
ode=y[x]*D[y[x],x]-y[x]==4/9*x+2*A*x^2+2*A^2*x^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\sqrt [4]{\frac {(-9 A y(x)+3 A x+1)^2}{(3 A x+1)^4}-1} \left (\frac {(-9 A y(x)+3 A x+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(3 A x-9 A y(x)+1)^2}{(3 A x+1)^4}\right )}{2 \sqrt [4]{3} (3 A x+1) \sqrt {(3 A x+1)^2} \sqrt [4]{\frac {A \left (6 (3 A x+1) y(x)-27 A y(x)^2+x (3 A x+2) (3 A x+1)^2\right )}{(3 A x+1)^4}}}+\sqrt {(3 A x+1)^2}\right )+c_1=0,y(x)\right ]
\]
2.22.11.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-2*A**2*x**3 - 2*A*x**2 - 4*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