55.22.26 problem 26
Internal
problem
ID
[13567]
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
:
26
Date
solved
:
Sunday, October 12, 2025 at 03:48:32 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=-\frac {2 x}{9}+\frac {A}{\sqrt {x}} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 134
ode:=y(x)*diff(y(x),x)-y(x) = -2/9*x+A/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 \,3^{{5}/{6}} 2^{{1}/{3}} \left (-2 x^{{3}/{2}}+9 A \right )}{\sqrt {x}\, \left (\left (27 \tan \left (\operatorname {RootOf}\left (18 \,3^{{5}/{6}} 2^{{1}/{3}} \int \frac {\left (\frac {A}{x^{{3}/{2}}}\right )^{{2}/{3}} \sqrt {x}}{-2 x^{{3}/{2}}+9 A}d x +\ln \left (-8 \sqrt {3}\, \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )^{3}-8 \cos \left (\textit {\_Z} \right )^{4}-4 \sqrt {3}\, \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )+16 \cos \left (\textit {\_Z} \right )^{2}+1\right ) \sqrt {3}-12 \sqrt {3}\, c_1 -12 \textit {\_Z} \right )\right )-9 \sqrt {3}\right ) \left (\frac {A}{x^{{3}/{2}}}\right )^{{1}/{3}}-6 \,2^{{1}/{3}} 3^{{5}/{6}}\right )}
\]
✓ Mathematica. Time used: 0.467 (sec). Leaf size: 282
ode=y[x]*D[y[x],x]-y[x]==-2/9*x+A*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\log \left (9 A^{2/3}+3 \sqrt [3]{6} \sqrt [3]{A} \sqrt {x}+6^{2/3} x\right )+2 \sqrt {3} \arctan \left (\frac {-\frac {6 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}-27}{27 \sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{6} \sqrt {x}}{\sqrt [3]{A}}+3}{3 \sqrt {3}}\right )+2 \log \left (\frac {1}{27} \left (27-\frac {3 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}\right )\right )=\log \left (\frac {1}{81} \left (\frac {6^{2/3} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )^2}{A^{2/3} y(x)^2}+\frac {9 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}+81\right )\right )+2 \log \left (3 \sqrt [3]{A}-\sqrt [3]{6} \sqrt {x}\right )+6 c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/sqrt(x) + 2*x/9 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out