15.8.19 problem 19

Internal problem ID [3022]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 19
Date solved : Sunday, March 30, 2025 at 01:10:46 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 0.442 (sec). Leaf size: 80

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

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

✓ Sympy. Time used: 1.623 (sec). Leaf size: 73

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

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