32.1.1 problem First order with homogeneous Coefficients. Exercise 7.2, page 61
Internal
problem
ID
[5771]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
7
Problem
number
:
First
order
with
homogeneous
Coefficients.
Exercise
7.2,
page
61
Date
solved
:
Sunday, March 30, 2025 at 10:14:12 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 x y+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.019 (sec). Leaf size: 205
ode:=2*x*y(x)+(x^2+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2 \left (c_1 \,x^{2}-\frac {\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4}\right )}{\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {x^{2} \sqrt {c_1}\, \left (i \sqrt {3}-1\right )}{\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}+i \sqrt {3}\, \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}+4 c_1 \,x^{2}-\left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 x^{6} c_1^{3}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Mathematica. Time used: 15.736 (sec). Leaf size: 401
ode=2*x*y[x]+(x^2+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]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2}{4 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
y(x)\to 0 \\
y(x)\to \frac {1}{2} \sqrt [6]{x^6} \left (\frac {\left (1-i \sqrt {3}\right ) \left (x^6\right )^{2/3}}{x^4}-i \sqrt {3}-1\right ) \\
y(x)\to \frac {1}{2} \sqrt [6]{x^6} \left (\frac {\left (1+i \sqrt {3}\right ) \left (x^6\right )^{2/3}}{x^4}+i \sqrt {3}-1\right ) \\
y(x)\to \sqrt [6]{x^6}-\frac {\left (x^6\right )^{5/6}}{x^4} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x) + (x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out