30.5.15 problem 15

Internal problem ID [7514]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 04:41:11 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} y^{\prime }&=\frac {x^{2}-y^{2}}{3 x y} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 43

ode:=diff(y(x),x) = 1/3*(x^2-y(x)^2)/x/y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {x^{{2}/{3}} \left (x^{{8}/{3}}+4 c_1 \right )}}{2 x^{{2}/{3}}} \\ y &= \frac {\sqrt {x^{{2}/{3}} \left (x^{{8}/{3}}+4 c_1 \right )}}{2 x^{{2}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 3.519 (sec). Leaf size: 50

ode=D[y[x],x]==( x^2-y[x]^2 )/(3*x*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} \sqrt {x^2+\frac {4 c_1}{x^{2/3}}}\\ y(x)&\to \sqrt {\frac {x^2}{4}+\frac {c_1}{x^{2/3}}} \end{align*}

✓ Sympy. Time used: 0.294 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**2 - y(x)**2)/(3*x*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {\frac {C_{1}}{x^{\frac {2}{3}}} + x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {\frac {C_{1}}{x^{\frac {2}{3}}} + x^{2}}}{2}\right ] \]