15.2.16 problem 16
Internal
problem
ID
[2886]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
6,
page
25
Problem
number
:
16
Date
solved
:
Tuesday, March 04, 2025 at 02:59:02 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }+1&=0 \end{align*}
✓ Maple. Time used: 0.494 (sec). Leaf size: 221
ode:=(x/y(x)+y(x)/x)*diff(y(x),x)+1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \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 &= \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 &= \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 &= -\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: 0.102 (sec). Leaf size: 121
ode=(x/y[x]+y[x]/x)*D[y[x],x]+1==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}}} \\
\end{align*}
✓ Sympy. Time used: 3.618 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x/y(x) + y(x)/x)*Derivative(y(x), x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ]
\]