78.8.36 problem 36
Internal
problem
ID
[18155]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Miscellaneous
Problems
for
Chapter
2.
Problems
at
page
99
Problem
number
:
36
Date
solved
:
Thursday, March 13, 2025 at 11:45:28 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 3 x^{2} y-y^{3}-\left (3 x y^{2}-x^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.023 (sec). Leaf size: 244
ode:=3*x^2*y(x)-y(x)^3-(3*x*y(x)^2-x^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x^{4} c_{1}^{2} 12^{{1}/{3}}+{\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}\right )}{6 c_{1} x {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-1-i \sqrt {3}\right ) {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}+c_{1}^{2} 2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{4}\right )}{12 {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}} c_{1} x} \\
y &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_{1}^{2} 2^{{2}/{3}} x^{4}\right ) 3^{{1}/{3}}}{12 {\left (\left (9+\sqrt {-12 c_{1}^{4} x^{8}+81}\right ) x^{2} c_{1} \right )}^{{1}/{3}} c_{1} x} \\
\end{align*}
✓ Mathematica. Time used: 60.22 (sec). Leaf size: 338
ode=(3*x^2*y[x]-y[x]^3) - (3*x*y[x]^2-x^3) * 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 {i \left (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}+\sqrt [3]{2} \sqrt [6]{3} \left (-6-2 i \sqrt {3}\right ) x^2\right )}{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