61.22.54 problem 54
Internal
problem
ID
[12300]
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
:
54
Date
solved
:
Monday, March 31, 2025 at 05:05:19 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=6 x +\frac {A}{x^{4}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 217
ode:=y(x)*diff(y(x),x)-y(x) = 6*x+A/x^4;
dsolve(ode,y(x), singsol=all);
\[
c_1 +\frac {5 \,5^{{2}/{3}} \left (x +\frac {y}{2}\right ) \left (x 3^{{5}/{6}} A \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {2 A}{3 x^{3} \left (2 x +y\right )^{2}}\right ) \left (\frac {12 x^{5}+12 y x^{4}+3 x^{3} y^{2}+2 A}{x^{9} \left (2 x +y\right )^{6}}\right )^{{1}/{6}}+\frac {24 \left (6 x^{5}+6 y x^{4}+\frac {3 x^{3} y^{2}}{2}+A \right ) \left (\frac {\left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{2}/{3}} y}{6}+x^{{5}/{2}} \left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{5}/{3}} \left (x +\frac {y}{2}\right )\right )}{5}\right )}{2 x^{{11}/{2}} \left (\frac {12 x^{5}+12 y x^{4}+3 x^{3} y^{2}+2 A}{x^{3} \left (2 x +y\right )^{2}}\right )^{{1}/{6}} \left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{7}/{3}} \left (2 x +y\right )^{4}} = 0
\]
✓ Mathematica. Time used: 1.286 (sec). Leaf size: 213
ode=y[x]*D[y[x],x]-y[x]==6*x+A*x^(-4);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [c_1=\frac {i \left (-\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6} \left (-10\ 2^{5/6} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )-5\ 2^{5/6} x^4 y(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )+A \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt {A} x^{5/2} \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}},y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/x**4 - 6*x + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -A/(x**4*y(x)) - 6*x/y(x) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method