78.3.18 problem 6 (a)

Internal problem ID [18050]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 7 (Homogeneous Equations). Problems at page 67
Problem number : 6 (a)
Date solved : Monday, March 31, 2025 at 05:00:05 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} y^{\prime }&=\frac {1-x y^{2}}{2 x^{2} y} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

ode:=diff(y(x),x) = 1/2*(1-x*y(x)^2)/x^2/y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.184 (sec). Leaf size: 40

ode=D[y[x],x]==(1-x*y[x]^2)/(2*x^2*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.346 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (-x*y(x)**2 + 1)/(2*x**2*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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