86.2.14 problem 14

Internal problem ID [23088]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3c at page 50
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:19:51 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} y^{\prime }&=\frac {y}{x}-\frac {x}{y} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(y(x),x) = y(x)/x-x/y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-2 \ln \left (x \right )+c_1}\, x \\ y &= -\sqrt {-2 \ln \left (x \right )+c_1}\, x \\ \end{align*}

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 36

ode=D[y[x],x]==y[x]/x - x/y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \sqrt {-2 \log (x)+c_1}\\ y(x)&\to x \sqrt {-2 \log (x)+c_1} \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x/y(x) + Derivative(y(x), x) - y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - x \sqrt {C_{1} - 2 \log {\left (x \right )}}, \ y{\left (x \right )} = x \sqrt {C_{1} - 2 \log {\left (x \right )}}\right ] \]