25.3.1 problem 2

Internal problem ID [6866]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 4
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 03:58:47 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 5.802 (sec). Leaf size: 70

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

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

✓ Sympy. Time used: 0.764 (sec). Leaf size: 20

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

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