49.22.1 problem 1(a)
Internal
problem
ID
[7747]
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
:
1(a)
Date
solved
:
Sunday, March 30, 2025 at 12:22:33 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 x y+\left (x^{2}+3 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 185
ode:=2*x*y(x)+(x^2+3*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {-12 c_1 \,x^{2}+\left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{2}/{3}}}{6 \left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{1}/{3}}}{12 \sqrt {c_1}}-\frac {\sqrt {c_1}\, x^{2} \left (i \sqrt {3}-1\right )}{\left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{1}/{3}}}{12 \sqrt {c_1}}+\frac {\sqrt {c_1}\, x^{2} \left (1+i \sqrt {3}\right )}{\left (108+12 \sqrt {12 x^{6} c_1^{3}+81}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 23.825 (sec). Leaf size: 441
ode=2*x*y[x]+(x^2+3*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 (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\
y(x)\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{12 \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\
y(x)\to \frac {2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^2}{12 \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\
y(x)\to 0 \\
y(x)\to \frac {\sqrt [3]{x^6}-x^2}{\sqrt {3} \sqrt [6]{x^6}} \\
y(x)\to \frac {\left (\sqrt {3}+3 i\right ) x^2-\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^6}}{6 \sqrt [6]{x^6}} \\
y(x)\to \frac {1}{6} \sqrt [6]{x^6} \left (\frac {\left (\sqrt {3}-3 i\right ) \left (x^6\right )^{2/3}}{x^4}-\sqrt {3}-3 i\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x) + (x**2 + 3*y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out