79.1.23 problem 4 (v)
Internal
problem
ID
[18431]
Book
:
Elementary
Differential
Equations.
By
R.L.E.
Schwarzenberger.
Chapman
and
Hall.
London.
First
Edition
(1969)
Section
:
Chapter
3.
Solutions
of
first-order
equations.
Exercises
at
page
47
Problem
number
:
4
(v)
Date
solved
:
Thursday, March 13, 2025 at 11:56:55 AM
CAS
classification
:
[_separable]
\begin{align*} x^{\prime }+2 t x+t x^{4}&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 121
ode:=diff(x(t),t)+2*t*x(t)+t*x(t)^4 = 0;
dsolve(ode,x(t), singsol=all);
\begin{align*}
x &= \frac {2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_{1} -1\right )^{2}\right )}^{{1}/{3}}}{2 \,{\mathrm e}^{3 t^{2}} c_{1} -1} \\
x &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_{1} -1\right )^{2}\right )}^{{1}/{3}}}{4 \,{\mathrm e}^{3 t^{2}} c_{1} -2} \\
x &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} {\left (\left (2 \,{\mathrm e}^{3 t^{2}} c_{1} -1\right )^{2}\right )}^{{1}/{3}}}{4 \,{\mathrm e}^{3 t^{2}} c_{1} -2} \\
\end{align*}
✓ Mathematica. Time used: 10.919 (sec). Leaf size: 177
ode=D[x[t],t]+2*t*x[t]+t*x[t]^4==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -\frac {\sqrt [3]{-2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}} \\
x(t)\to \frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}} \\
x(t)\to \frac {(-1)^{2/3} \sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{e^{3 t^2}-e^{6 c_1}}} \\
x(t)\to 0 \\
x(t)\to \sqrt [3]{-2} \\
x(t)\to -\sqrt [3]{2} \\
x(t)\to -(-1)^{2/3} \sqrt [3]{2} \\
x(t)\to \frac {1-i \sqrt {3}}{2^{2/3}} \\
\end{align*}
✓ Sympy. Time used: 3.096 (sec). Leaf size: 88
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(t*x(t)**4 + 2*t*x(t) + Derivative(x(t), t),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
\left [ x{\left (t \right )} = \sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}}, \ x{\left (t \right )} = \frac {\sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}} \left (-1 - \sqrt {3} i\right )}{2}, \ x{\left (t \right )} = \frac {\sqrt [3]{2} \sqrt [3]{- \frac {C_{1}}{C_{1} - e^{3 t^{2}}}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]