79.5.2 problem (b)

Internal problem ID [21094]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises IX at page 45
Problem number : (b)
Date solved : Thursday, October 02, 2025 at 07:08:08 PM
CAS classification : [_rational]

\begin{align*} x y^{2}-y^{3}+\left (1-x y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 73

ode:=x*y(x)^2-y(x)^3+(1-x*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{2}+2 c_1 +\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}-16 x}}{4 x} \\ y &= \frac {x^{2}+2 c_1 -\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}-16 x}}{4 x} \\ \end{align*}

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 94

ode=(x*y[x]^2-y[x]^3)+(1-x*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2-\sqrt {x^4+4 c_1 x^2-16 x+4 c_1{}^2}+2 c_1}{4 x}\\ y(x)&\to \frac {x^2+\sqrt {x^4+4 c_1 x^2-16 x+4 c_1{}^2}+2 c_1}{4 x}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 2.068 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**2 + (-x*y(x)**2 + 1)*Derivative(y(x), x) - y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {2 C_{1} + x^{2} - \sqrt {4 C_{1}^{2} + 4 C_{1} x^{2} + x^{4} - 16 x}}{4 x} \]