89.6.12 problem 12
Internal
problem
ID
[24395]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
2.
Equations
of
the
first
order
and
first
degree.
Miscellaneous
Exercises
at
page
45
Problem
number
:
12
Date
solved
:
Thursday, October 02, 2025 at 10:24:27 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (2 x^{3}-x^{2} y+y^{3}\right ) y-x \left (2 x^{3}+y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 283
ode:=y(x)*(2*x^3-x^2*y(x)+y(x)^3)-x*(2*x^3+y(x)^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{2}/{3}}-6 \ln \left (x \right )-6 c_1 \right ) x}{3 \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{1}/{3}}} \\
y &= -\frac {x \left (\left (\frac {i \sqrt {3}}{6}+\frac {1}{6}\right ) \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{2}/{3}}+\left (\ln \left (x \right )+c_1 \right ) \left (i \sqrt {3}-1\right )\right )}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{1}/{3}}} \\
y &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{2}/{3}}}{6}+\left (\ln \left (x \right )+c_1 \right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} c_1 +18 \ln \left (x \right ) c_1^{2}+6 c_1^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 48.873 (sec). Leaf size: 362
ode=y[x]*( 2*x^3-x^2*y[x]+y[x]^3) - x*( 2*x^3+y[x]^3) *D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-6^{2/3} x^2 \log (x)+6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}\right ){}^{2/3}}{3 \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) x^2 (\log (x)-c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) x^2 (-\log (x)+c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(2*x**3 + y(x)**3)*Derivative(y(x), x) + (2*x**3 - x**2*y(x) + y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out