84.6.5 problem 3.13

Internal problem ID [22106]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 3. Classification of first-order differential equations. Supplementary problems
Problem number : 3.13
Date solved : Thursday, October 02, 2025 at 08:25:02 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{\prime }&=\frac {x y^{2}}{x^{2} y+y^{3}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 6.217 (sec). Leaf size: 49

ode=D[y[x],x]==(x*y[x]^2)/(x^2*y[x]+y[x]^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.713 (sec). Leaf size: 17

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

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