61.22.18 problem 18
Internal
problem
ID
[12264]
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
:
18
Date
solved
:
Monday, March 31, 2025 at 04:57:12 AM
CAS
classification
:
[[_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 710
ode:=y(x)*diff(y(x),x)-y(x) = 2*a^2/(8*a^2+x^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {512 \,{\mathrm e}^{-\frac {\left (x -y\right )^{2} \left (-64 \sqrt {8 a^{2}+x^{2}}\, a^{6}-108 \sqrt {8 a^{2}+x^{2}}\, a^{4} x^{2}-25 a^{2} \sqrt {8 a^{2}+x^{2}}\, x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{6}+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )^{2}}{2 \left (128 a^{6}-66 x \sqrt {8 a^{2}+x^{2}}\, a^{4}+150 a^{4} x^{2}-23 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{3}+27 a^{2} x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{5}+x^{6}\right )^{2} a^{2} \left (-x \sqrt {8 a^{2}+x^{2}}+4 a^{2}+x^{2}\right )}} a \left (-\frac {33 x \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) \sqrt {8 a^{2}+x^{2}}}{64}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right )+128 \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x -y\right ) \left (\sqrt {8 a^{2}+x^{2}}\, \left (-64 a^{6}-108 a^{4} x^{2}-25 a^{2} x^{4}-x^{6}\right )+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )}{2 \sqrt {-x \sqrt {8 a^{2}+x^{2}}+4 a^{2}+x^{2}}\, \left (\left (-66 a^{5} x -23 a^{3} x^{3}-a \,x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+128 a^{7}+150 a^{5} x^{2}+27 a^{3} x^{4}+a \,x^{6}\right )}\right ) \sqrt {2}\, \sqrt {\pi }+c_1 \right ) \sqrt {-x \sqrt {8 a^{2}+x^{2}}+4 a^{2}+x^{2}}\, \left (\frac {\left (\left (a^{4}+\frac {21}{32} a^{2} x^{2}+\frac {1}{32} x^{4}\right ) y-\frac {33 x \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right )}{16}\right ) \sqrt {8 a^{2}+x^{2}}}{4}+\frac {\left (-25 a^{4} x -\frac {25}{4} a^{2} x^{3}-\frac {1}{4} x^{5}\right ) y}{32}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right )}{\sqrt {-x \sqrt {8 a^{2}+x^{2}}+4 a^{2}+x^{2}}\, \left (\left (\left (32 a^{4}+21 a^{2} x^{2}+x^{4}\right ) y-66 a^{4} x -23 a^{2} x^{3}-x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+\left (-100 a^{4} x -25 a^{2} x^{3}-x^{5}\right ) y+128 a^{6}+150 a^{4} x^{2}+27 a^{2} x^{4}+x^{6}\right )} = 0
\]
✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==2*a^2/Sqrt[x^2+8*a^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-2*a**2/sqrt(8*a**2 + x**2) + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -2*a**2/(sqrt(8*a**2 + x**2)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method