32.1.5 problem First order with homogeneous Coefficients. Exercise 7.6, page 61

Internal problem ID [5775]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.6, page 61
Date solved : Tuesday, March 04, 2025 at 11:41:59 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 22

ode:=2*x^2*y(x)+y(x)^3+(x*y(x)^2-2*x^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \sqrt {2}\, \sqrt {-\frac {1}{\operatorname {LambertW}\left (-2 c_{1} x^{4}\right )}}\, x \]

✓ Mathematica. Time used: 6.262 (sec). Leaf size: 70

ode=(2*x^2*y[x]+y[x]^3)+(x*y[x]^2-2*x^3)*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^4\right )}} \\ y(x)\to \frac {i \sqrt {2} x}{\sqrt {W\left (-2 e^{-3-2 c_1} x^4\right )}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.836 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x) + (-2*x**3 + x*y(x)**2)*Derivative(y(x), x) + y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{2 C_{1} + \frac {W\left (- 2 x^{4} e^{- 4 C_{1}}\right )}{2}}}{x} \]