89.4.2 problem 2
Internal
problem
ID
[24324]
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.
Exercises
at
page
39
Problem
number
:
2
Date
solved
:
Thursday, October 02, 2025 at 10:18:05 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y \left (y^{3}-x \right )+x \left (y^{3}+x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 342
ode:=y(x)*(y(x)^3-x)+x*(y(x)^3+x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}+\frac {c_1^{2}}{\left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}}+c_1}{6 x} \\
y &= \frac {i \sqrt {3}\, c_1^{2}-i \left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{2}/{3}} \sqrt {3}-c_1^{2}+2 c_1 \left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}-\left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{2}/{3}}}{12 x \left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}} \\
y &= \frac {\left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )-\frac {c_1 \left (i c_1 \sqrt {3}+c_1 -2 \left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}\right )}{\left (6 x^{2} \sqrt {81 x^{4}+3 c_1^{3}}+54 x^{4}+c_1^{3}\right )^{{1}/{3}}}}{12 x} \\
\end{align*}
✓ Mathematica. Time used: 16.569 (sec). Leaf size: 357
ode=y[x]*( y[x]^3-x)+x*(y[x]^3+x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\frac {2\ 2^{2/3} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}+\sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}+2 c_1}{6 x}\\ y(x)&\to \frac {-\frac {2 i 2^{2/3} \left (\sqrt {3}-i\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to \frac {\frac {2 i 2^{2/3} \left (\sqrt {3}+i\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x + y(x)**3)*Derivative(y(x), x) + (-x + y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out