61.22.18 problem 18
Internal
problem
ID
[12343]
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
:
Tuesday, January 28, 2025 at 07:55:04 PM
CAS
classification
:
[[_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime } y-y&=\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 715
dsolve(y(x)*diff(y(x),x)-y(x)=2*a^2/sqrt(x^2+8*a^2),y(x), singsol=all)
\[
\frac {512 \left (-\frac {33 x \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) \sqrt {8 a^{2}+x^{2}}}{64}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 x^{4} a^{2}}{128}+\frac {x^{6}}{128}\right ) a \,{\mathrm e}^{-\frac {\left (x -y\right )^{2} \left (-64 \sqrt {8 a^{2}+x^{2}}\, a^{6}-108 \sqrt {8 a^{2}+x^{2}}\, a^{4} x^{2}-25 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{6}+328 x \,a^{6}+200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )^{2}}{2 \left (128 a^{6}+150 a^{4} x^{2}-66 \sqrt {8 a^{2}+x^{2}}\, a^{4} x +27 x^{4} a^{2}-23 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{3}+x^{6}-\sqrt {8 a^{2}+x^{2}}\, x^{5}\right )^{2} a^{2} \left (-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}\right )}}-128 \left (\frac {\left (\left (a^{4}+\frac {21}{32} a^{2} x^{2}+\frac {1}{32} x^{4}\right ) y-\frac {33 x \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right )}{16}\right ) \sqrt {8 a^{2}+x^{2}}}{4}+\frac {\left (-25 a^{4} x -\frac {25}{4} a^{2} x^{3}-\frac {1}{4} x^{5}\right ) y}{32}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 x^{4} a^{2}}{128}+\frac {x^{6}}{128}\right ) \left (\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\left (x -y\right ) \sqrt {2}\, \left (-\left (-64 a^{6}-108 a^{4} x^{2}-25 x^{4} a^{2}-x^{6}\right ) \sqrt {8 a^{2}+x^{2}}-328 x \,a^{6}-200 a^{4} x^{3}-29 a^{2} x^{5}-x^{7}\right )}{2 \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-66 x \,a^{5}-23 a^{3} x^{3}-a \,x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+128 a^{7}+150 a^{5} x^{2}+27 a^{3} x^{4}+a \,x^{6}\right )}\right )-c_{1} \right ) \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}}{\sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (\left (32 a^{4}+21 a^{2} x^{2}+x^{4}\right ) y-66 a^{4} x -23 a^{2} x^{3}-x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+\left (-100 a^{4} x -25 a^{2} x^{3}-x^{5}\right ) y+128 a^{6}+150 a^{4} x^{2}+27 x^{4} a^{2}+x^{6}\right )} = 0
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[y[x]*D[y[x],x]-y[x]==2*a^2/Sqrt[x^2+8*a^2],y[x],x,IncludeSingularSolutions -> True]
Not solved