71.8.34 problem 12 (a)

Internal problem ID [14397]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 12 (a)
Date solved : Monday, March 31, 2025 at 12:24:59 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.692 (sec). Leaf size: 16

ode:=diff(y(x),x) = x*y(x)/(x^2+y(x)^2); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 7.124 (sec). Leaf size: 15

ode=D[y[x],x]==x*y[x]/(x^2+y[x]^2); 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x}{\sqrt {W\left (x^2\right )}} \]

✓ Sympy. Time used: 1.068 (sec). Leaf size: 10

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 = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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