34.2.3 problem 3

Internal problem ID [6033]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 3
Date solved : Sunday, March 30, 2025 at 10:36:29 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 32

ode:=y(x)^2+x*y(x)^2+(x^2-x^2*y(x))*diff(y(x),x) = 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: 5.252 (sec). Leaf size: 30

ode=(y[x]^2+x*y[x]^2)+(x^2-y[x]*x^2)*D[y[x],x]==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: 0.982 (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)
 

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