76.5.14 problem 14
Internal
problem
ID
[17363]
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
:
Monday, March 31, 2025 at 04:07:46 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: 9.983 (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: 2.942 (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)
\[
\left [ y{\left (t \right )} = \sqrt [3]{3} \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}}{2}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{\frac {1}{C_{1} e^{3 t} + 3 t + 1}}}{2}\right ]
\]