68.6.11 problem 12

Internal problem ID [17321]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 12
Date solved : Thursday, October 02, 2025 at 02:02:30 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 56

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

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

✓ Sympy. Time used: 0.662 (sec). Leaf size: 48

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

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