89.10.2 problem 2
Internal
problem
ID
[24495]
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
77
Problem
number
:
2
Date
solved
:
Thursday, October 02, 2025 at 10:42:44 PM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\begin{align*} y^{3}+y+1+x \left (x -3 y^{2}-1\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 286
ode:=y(x)^3+y(x)+1+x*(x-3*y(x)^2-1)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{2}/{3}}+12 x -12}{6 \left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{1}/{3}}} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{1}/{3}}}{12}+\frac {\left (i \sqrt {3}-1\right ) \left (x -1\right )}{\left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{2}/{3}}}{12}+\left (-i \sqrt {3}-1\right ) \left (x -1\right )}{\left (-108+108 c_1 x +12 \sqrt {93-12 x^{3}+\left (81 c_1^{2}+36\right ) x^{2}+\left (-162 c_1 -36\right ) x}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 51.768 (sec). Leaf size: 381
ode=( y[x]^3+y[x]+1 )+x*( x-3*y[x]^2-1 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{2} (3-3 x)}{3 \sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}-\frac {\sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}{3 \sqrt [3]{2}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}{6 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) (3-3 x)}{3\ 2^{2/3} \sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}{6 \sqrt [3]{2}}-\frac {\left (1-i \sqrt {3}\right ) (3-3 x)}{3\ 2^{2/3} \sqrt [3]{27 c_1 x+\sqrt {4 (3-3 x)^3+(27+27 c_1 x){}^2}+27}}\\ y(x)&\to \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x - 3*y(x)**2 - 1)*Derivative(y(x), x) + y(x)**3 + y(x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out