66.1.17 problem Problem 17
Internal
problem
ID
[13793]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
1,
First-Order
Differential
Equations.
Problems
page
88
Problem
number
:
Problem
17
Date
solved
:
Monday, March 31, 2025 at 08:12:20 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y^{\prime }&=\frac {y}{x +y^{3}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 220
ode:=diff(y(x),x) = y(x)/(x+y(x)^3);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_1}{3 \left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {i \left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{2}/{3}} \sqrt {3}+6 i \sqrt {3}\, c_1 +\left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_1}{6 \left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{2}/{3}} \sqrt {3}+6 i \sqrt {3}\, c_1 -\left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{2}/{3}}+6 c_1}{6 \left (27 x +3 \sqrt {24 c_1^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 1.855 (sec). Leaf size: 263
ode=D[y[x],x]==y[x]/(x+y[x]^3);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2\ 3^{2/3} c_1-\sqrt [3]{3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}}{3 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to \frac {\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - y(x)/(x + y(x)**3),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out