71.4.2 problem 2

Internal problem ID [14303]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number : 2
Date solved : Monday, March 31, 2025 at 12:16:19 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 7.033 (sec). Leaf size: 49

ode=D[y[x],x]==x*y[x]/( x^2+y[x]^2); 
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: 1.098 (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)
 

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