89.4.3 problem 3

Internal problem ID [24325]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:18:08 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} x^{3} y^{3}+1+x^{4} y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 64

ode:=x^3*y(x)^3+1+x^4*y(x)^2*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 69

ode=( x^3*y[x]^3+1)+(x^4*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.953 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*y(x)**2*Derivative(y(x), x) + x**3*y(x)**3 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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