35.4.9 problem 9

Internal problem ID [6127]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 4. OTHER METHODS FOR FIRST-ORDER EQUATIONS. page 406
Problem number : 9
Date solved : Sunday, March 30, 2025 at 10:40:31 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 19

ode:=x*y(x)+(y(x)^2-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: 7.968 (sec). Leaf size: 60

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

\begin{align*} y(x)\to -\frac {i x}{\sqrt {W\left (-e^{-3-2 c_1} x^2\right )}} \\ y(x)\to \frac {i x}{\sqrt {W\left (-e^{-3-2 c_1} x^2\right )}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.277 (sec). Leaf size: 19

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)
 

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