89.7.13 problem 13
Internal
problem
ID
[24444]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
4.
Additional
topics
on
equations
of
first
order
and
first
degree.
Exercises
at
page
61
Problem
number
:
13
Date
solved
:
Thursday, October 02, 2025 at 10:35:10 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y+x \left (3 y x -2\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.666 (sec). Leaf size: 28
ode:=y(x)+x*(3*x*y(x)-2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\operatorname {RootOf}\left (c_1 \,\textit {\_Z}^{9}-c_1 \,\textit {\_Z}^{6}-x^{3}\right )^{3}}{x}
\]
✓ Mathematica. Time used: 45.674 (sec). Leaf size: 500
ode=y[x]+x*(3*x*y[x]-2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2^{2/3} e^{-c_1} \sqrt [3]{-27 e^{2 c_1} x^3+3 \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+2 e^{3 c_1}}+\frac {2 e^{c_1}}{\sqrt [3]{-\frac {27}{2} e^{2 c_1} x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+e^{3 c_1}}}+2}{6 x}\\ y(x)&\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) e^{-c_1} \sqrt [3]{-27 e^{2 c_1} x^3+3 \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+2 e^{3 c_1}}-\frac {2 i \left (\sqrt {3}-i\right ) e^{c_1}}{\sqrt [3]{-\frac {27}{2} e^{2 c_1} x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+e^{3 c_1}}}+4}{12 x}\\ y(x)&\to \frac {-i 2^{2/3} \left (\sqrt {3}-i\right ) e^{-c_1} \sqrt [3]{-27 e^{2 c_1} x^3+3 \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+2 e^{3 c_1}}+\frac {2 i \left (\sqrt {3}+i\right ) e^{c_1}}{\sqrt [3]{-\frac {27}{2} e^{2 c_1} x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{4 c_1} x^3 \left (27 x^3-4 e^{c_1}\right )}+e^{3 c_1}}}+4}{12 x}\\ y(x)&\to 0\\ y(x)&\to \frac {1}{x} \end{align*}
✓ Sympy. Time used: 33.051 (sec). Leaf size: 388
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(3*x*y(x) - 2)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {- 2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}} + \frac {2}{x} - \frac {2 \sqrt {3} i}{x} + \frac {4 \sqrt [3]{2}}{x^{2} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {- 2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}} + 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}} + \frac {2}{x} + \frac {2 \sqrt {3} i}{x} + \frac {4 \sqrt [3]{2}}{x^{2} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}}}}{6 \left (1 + \sqrt {3} i\right )}, \ y{\left (x \right )} = - \frac {2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}}}{6} + \frac {1}{3 x} - \frac {\sqrt [3]{2}}{3 x^{2} \sqrt [3]{- 27 C_{1} + 3 \sqrt {3} \sqrt {C_{1} \left (27 C_{1} + \frac {4}{x^{3}}\right )} - \frac {2}{x^{3}}}}\right ]
\]