49.22.9 problem 2(a)
Internal
problem
ID
[7755]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
5.
Existence
and
uniqueness
of
solutions
to
first
order
equations.
Page
198
Problem
number
:
2(a)
Date
solved
:
Sunday, March 30, 2025 at 12:22:44 PM
CAS
classification
:
[_separable]
\begin{align*} 2 y^{3}+2+3 x y^{2} y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 71
ode:=2*y(x)^3+2+3*x*y(x)^2*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\left (\left (-x^{2}+c_1 \right ) x \right )}^{{1}/{3}}}{x} \\
y &= -\frac {{\left (\left (-x^{2}+c_1 \right ) x \right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 x} \\
y &= \frac {{\left (\left (-x^{2}+c_1 \right ) x \right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 x} \\
\end{align*}
✓ Mathematica. Time used: 0.301 (sec). Leaf size: 215
ode=(3*y[x]^3+2)+(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 {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{-2 x^3+e^{9 c_1}}}{x} \\
y(x)\to \frac {\sqrt [3]{-2 x^3+e^{9 c_1}}}{\sqrt [3]{3} x} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-2 x^3+e^{9 c_1}}}{\sqrt [3]{3} x} \\
y(x)\to \sqrt [3]{-\frac {2}{3}} \\
y(x)\to -\sqrt [3]{\frac {2}{3}} \\
y(x)\to -(-1)^{2/3} \sqrt [3]{\frac {2}{3}} \\
y(x)\to \frac {\sqrt [3]{-\frac {2}{3}} x^2}{\left (-x^3\right )^{2/3}} \\
y(x)\to \frac {\sqrt [3]{\frac {2}{3}} \sqrt [3]{-x^3}}{x} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} \sqrt [3]{-x^3}}{x} \\
\end{align*}
✓ Sympy. Time used: 1.678 (sec). Leaf size: 78
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x*y(x)**2*Derivative(y(x), x) + 2*y(x)**3 + 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (- \sqrt [3]{-1} - \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) \sqrt [3]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{-1} + \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) \sqrt [3]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {C_{1}}{x^{2}} - 1}\right ]
\]