6.5.45 problem 44

Internal problem ID [1669]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 44
Date solved : Tuesday, September 30, 2025 at 04:50:42 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 69

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

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

✓ Sympy. Time used: 0.871 (sec). Leaf size: 58

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

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