76.12.5 problem Ex. 5

Internal problem ID [20055]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 5
Date solved : Thursday, October 02, 2025 at 05:17:17 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 4.38 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to -\frac {1}{W\left (-\frac {e^{\frac {1}{x}-c_1}}{x}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.581 (sec). Leaf size: 22

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

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