69.7.13 problem 188
Internal
problem
ID
[18093]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
7,
Total
differential
equations.
The
integrating
factor.
Exercises
page
61
Problem
number
:
188
Date
solved
:
Thursday, October 02, 2025 at 02:41:53 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 3 x^{2} y+y^{3}+\left (x^{3}+3 x y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 274
ode:=3*x^2*y(x)+y(x)^3+(x^3+3*x*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {12^{{1}/{3}} \left (c_1^{2} x^{4} 12^{{1}/{3}}-{\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}\right )}{6 c_1 x {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}}} \\
y &= -\frac {2^{{2}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}+2^{{2}/{3}} c_1^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{4}\right ) 3^{{1}/{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}} c_1 x} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_1^{2} x^{4}\right )}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}} c_1 x} \\
\end{align*}
✓ Mathematica. Time used: 60.135 (sec). Leaf size: 338
ode=( 3*x^2*y[x]+y[x]^3)+(x^3+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 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}}\\ y(x)&\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{12 \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}}\\ y(x)&\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^2-i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}}{12 \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x) + (x**3 + 3*x*y(x)**2)*Derivative(y(x), x) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out