55.22.11 problem 11
Internal
problem
ID
[13552]
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
:
11
Date
solved
:
Sunday, October 12, 2025 at 03:39:54 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 354
ode:=y(x)*diff(y(x),x)-y(x) = -2/9*x+6*A^2*(1+2*A/x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-108 A^{3}-54 A^{2} \sqrt {x}+2 x^{{3}/{2}}}{3 \,{\mathrm e}^{\operatorname {RootOf}\left (18 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+36 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+108 A^{2} c_1 \,{\mathrm e}^{\textit {\_Z}}+36 A^{2} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +3 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+18 A \sqrt {x}\, c_1 \,{\mathrm e}^{\textit {\_Z}}+6 A \,{\mathrm e}^{\textit {\_Z}} \sqrt {x}\, \textit {\_Z} +108 A^{2} {\mathrm e}^{\textit {\_Z}}-18 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )-2 x \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-6 c_1 x \,{\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} x +324 A^{2}+54 A \sqrt {x}-18 x \right )} A +9 A +3 \sqrt {x}}
\]
✓ Mathematica. Time used: 5.749 (sec). Leaf size: 488
ode=y[x]*D[y[x],x]-y[x]==-2/9*x+6*A^2*(1+2*A*x^(-1/2));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right ) \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right ) \left (-\frac {\left (3 \left (3 \sqrt [3]{A^3}+\sqrt {x}\right ) y(x)+2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right )\right )}{9 \sqrt [3]{A^3} y(x)}+\left (\frac {2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)}{9 \sqrt [3]{A^3} y(x)}+1\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right )\right )-3\right )}{6561 \left (\frac {\left (2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)\right )^3}{729 A^3 y(x)^3}+\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{9 \sqrt [3]{A^3}}-2\right )}=\frac {2^{2/3} \left (A^3\right )^{2/3} \left (2 \text {arctanh}\left (\frac {1}{3}-\frac {2 \sqrt {x}}{9 A}\right )+\frac {9 A}{3 A+\sqrt {x}}\right )}{9 A^2}+c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-6*A**2*(2*A/sqrt(x) + 1) + 2*x/9 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out