61.22.11 problem 11
Internal
problem
ID
[12336]
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
:
Tuesday, January 28, 2025 at 07:54:44 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime } y-y&=-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 354
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y(x), singsol=all)
\[
y = \frac {-108 A^{3}-54 A^{2} \sqrt {x}+2 x^{{3}/{2}}}{3 \,{\mathrm e}^{\operatorname {RootOf}\left (36 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+18 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+108 A^{2} c_{1} {\mathrm e}^{\textit {\_Z}}+36 A^{2} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+18 A \sqrt {x}\, c_{1} {\mathrm e}^{\textit {\_Z}}+6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +108 A^{2} {\mathrm e}^{\textit {\_Z}}-18 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} x \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )-6 c_{1} x \,{\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+324 A^{2}+54 A \sqrt {x}-18 x \right )} A +9 A +3 \sqrt {x}}
\]
✓ Solution by Mathematica
Time used: 0.406 (sec). Leaf size: 137
DSolve[y[x]*D[y[x],x]-y[x]==-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{-\frac {2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)}{9 \sqrt [3]{2} \sqrt [3]{A^3} y(x)}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]=\frac {2^{2/3} \left (A^3\right )^{2/3} \left (2 \text {arctanh}\left (\frac {1}{3}-\frac {2 \sqrt {x}}{9 A}\right )+\frac {9 A}{3 A+\sqrt {x}}\right )}{9 A^2}+c_1,y(x)\right ]
\]