28.1.12 problem 12
Internal
problem
ID
[4318]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
12
Date
solved
:
Tuesday, March 04, 2025 at 06:20:55 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x y+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.955 (sec). Leaf size: 221
ode:=x*y(x)+(x^2+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {\left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\
y \left (x \right ) &= \frac {\sqrt {\left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\
y \left (x \right ) &= \frac {\sqrt {\left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (-c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\
y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 8.613 (sec). Leaf size: 218
ode=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 -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\
y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\
y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\
y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\
y(x)\to 0 \\
y(x)\to -\sqrt {-\sqrt {x^4}-x^2} \\
y(x)\to \sqrt {-\sqrt {x^4}-x^2} \\
y(x)\to -\sqrt {\sqrt {x^4}-x^2} \\
y(x)\to \sqrt {\sqrt {x^4}-x^2} \\
\end{align*}
✓ Sympy. Time used: 3.603 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x) + (x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ]
\]