78.1.25 problem 14.b

Internal problem ID [20951]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 14.b
Date solved : Thursday, October 02, 2025 at 07:00:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y x -\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

ode:=x*y(x)-(x^2+y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {\frac {1}{\operatorname {LambertW}\left (c_1 \,x^{2}\right )}}\, x \]

✓ Mathematica. Time used: 5.52 (sec). Leaf size: 49

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 -\frac {x}{\sqrt {W\left (e^{-2 c_1} x^2\right )}}\\ y(x)&\to \frac {x}{\sqrt {W\left (e^{-2 c_1} x^2\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.708 (sec). Leaf size: 17

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)
 

\[ y{\left (x \right )} = e^{C_{1} + \frac {W\left (x^{2} e^{- 2 C_{1}}\right )}{2}} \]