83.10.3 problem 3

Internal problem ID [21972]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:20:53 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 66

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

\begin{align*} y &= -x +c_1 \\ y &= \left (-3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} x \\ y &= -\frac {\left (-3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2} \\ y &= \frac {\left (-3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.303 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to -x+c_1\\ y(x)&\to x \sqrt [3]{-3 \log (x)+c_1}\\ y(x)&\to -\sqrt [3]{-1} x \sqrt [3]{-3 \log (x)+c_1}\\ y(x)&\to (-1)^{2/3} x \sqrt [3]{-3 \log (x)+c_1} \end{align*}

✓ Sympy. Time used: 1.650 (sec). Leaf size: 73

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

\[ \left [ y{\left (x \right )} = C_{1} - x, \ y{\left (x \right )} = \sqrt [3]{x^{3} \left (C_{1} - 3 \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} - 3 \log {\left (x \right )}\right )} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} - 3 \log {\left (x \right )}\right )} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]