61.22.46 problem 46
Internal
problem
ID
[12371]
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
:
46
Date
solved
:
Tuesday, January 28, 2025 at 07:57:17 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime } y-y&=\frac {6}{25} x -A \,x^{2} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 180
dsolve(y(x)*diff(y(x),x)-y(x)=6/25*x-A*x^2,y(x), singsol=all)
\[
-\frac {125 \left (5^{{1}/{3}} 2^{{5}/{6}} {\left (-\frac {1250 A \left (\frac {3 y^{2} A}{2}+\left (-\frac {6 x A}{5}+\frac {36}{125}\right ) y+x \left (x A -\frac {6}{25}\right )^{2}\right )}{\left (50 x A -125 y A -12\right )^{2}}\right )}^{{1}/{6}} y A \sqrt {-25 x A +6}-\frac {4 \left (\int _{}^{\frac {2 \left (-25 x A +6\right )^{{3}/{2}}}{-50 x A +125 y A +12}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} +c_{1} \right ) \left (\frac {\left (-25 x A +6\right )^{{3}/{2}}}{-50 x A +125 y A +12}\right )^{{1}/{3}} \left (x A -\frac {5 y A}{2}-\frac {6}{25}\right )}{5}\right )}{\left (\frac {\left (-25 x A +6\right )^{{3}/{2}}}{-50 x A +125 y A +12}\right )^{{1}/{3}} \left (100 x A -250 y A -24\right )} = 0
\]
✓ Solution by Mathematica
Time used: 2.072 (sec). Leaf size: 189
DSolve[y[x]*D[y[x],x]-y[x]==6/25*x-A*x^2,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {\sqrt [3]{5} \sqrt [6]{-\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}} \left (\frac {(-125 A y(x)+50 A x-12) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (-50 A x+125 A y(x)+12)^2}{2 (25 A x-6)^3}\right )}{\sqrt [3]{10} \sqrt {18-75 A x} (25 A x-6) \sqrt [6]{\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}}}+\sqrt {1-\frac {25 A x}{6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ]
\]