70.5.14 problem 14

Internal problem ID [18721]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 14
Date solved : Thursday, October 02, 2025 at 03:27:27 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=y \left (t y^{3}-1\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 127

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

\begin{align*} y &= \frac {3^{{1}/{3}} {\left (\left (3 \,{\mathrm e}^{3 t} c_1 +3 t +1\right )^{2}\right )}^{{1}/{3}}}{3 \,{\mathrm e}^{3 t} c_1 +3 t +1} \\ y &= -\frac {{\left (\left (3 \,{\mathrm e}^{3 t} c_1 +3 t +1\right )^{2}\right )}^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )}{6 \,{\mathrm e}^{3 t} c_1 +6 t +2} \\ y &= \frac {{\left (\left (3 \,{\mathrm e}^{3 t} c_1 +3 t +1\right )^{2}\right )}^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )}{6 \,{\mathrm e}^{3 t} c_1 +6 t +2} \\ \end{align*}

✓ Mathematica. Time used: 8.173 (sec). Leaf size: 124

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

\begin{align*} y(t)&\to \frac {1}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}}\\ y(t)&\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}}\\ y(t)&\to \frac {(-1)^{2/3}}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}}\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 1.972 (sec). Leaf size: 90

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

\[ \left [ y{\left (t \right )} = \sqrt [3]{3} \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}}{2}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}}{2}\right ] \]