74.8.5 problem 5
Internal
problem
ID
[16070]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Review
exercises,
page
80
Problem
number
:
5
Date
solved
:
Monday, March 31, 2025 at 02:38:40 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{5 t}}{y^{4}} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 140
ode:=diff(y(t),t) = exp(5*t)/y(t)^4;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \left ({\mathrm e}^{5 t}+c_1 \right )^{{1}/{5}} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left ({\mathrm e}^{5 t}+c_1 \right )^{{1}/{5}}}{4} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) \left ({\mathrm e}^{5 t}+c_1 \right )^{{1}/{5}}}{4} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) \left ({\mathrm e}^{5 t}+c_1 \right )^{{1}/{5}}}{4} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left ({\mathrm e}^{5 t}+c_1 \right )^{{1}/{5}}}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.754 (sec). Leaf size: 117
ode=D[y[t],t]==Exp[5*t]/y[t]^4;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to \sqrt [5]{e^{5 t}+5 c_1} \\
y(t)\to -\sqrt [5]{-1} \sqrt [5]{e^{5 t}+5 c_1} \\
y(t)\to (-1)^{2/5} \sqrt [5]{e^{5 t}+5 c_1} \\
y(t)\to -(-1)^{3/5} \sqrt [5]{e^{5 t}+5 c_1} \\
y(t)\to (-1)^{4/5} \sqrt [5]{e^{5 t}+5 c_1} \\
\end{align*}
✓ Sympy. Time used: 3.367 (sec). Leaf size: 163
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(Derivative(y(t), t) - exp(5*t)/y(t)**4,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\left [ y{\left (t \right )} = \sqrt [5]{C_{1} + e^{5 t}}, \ y{\left (t \right )} = \frac {\sqrt [5]{C_{1} + e^{5 t}} \left (- \sqrt {5} - 1 - \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{4}, \ y{\left (t \right )} = \frac {\sqrt [5]{C_{1} + e^{5 t}} \left (- \sqrt {5} - 1 + \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{4}, \ y{\left (t \right )} = \frac {\sqrt [5]{C_{1} + e^{5 t}} \left (-1 + \sqrt {5} - \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{4}, \ y{\left (t \right )} = \frac {\sqrt [5]{C_{1} + e^{5 t}} \left (-1 + \sqrt {5} + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{4}\right ]
\]