76.24.1 problem 1
Internal
problem
ID
[17729]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
6.
Systems
of
First
Order
Linear
Equations.
Section
6.2
(Basic
Theory
of
First
Order
Linear
Systems).
Problems
at
page
398
Problem
number
:
1
Date
solved
:
Monday, March 31, 2025 at 04:26:07 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }+4 y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 188
ode:=diff(diff(diff(diff(y(t),t),t),t),t)+5*diff(diff(diff(y(t),t),t),t)+4*y(t) = 0;
dsolve(ode,y(t), singsol=all);
\[
y = c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{-\frac {4 t \left (\frac {\left (190+6 \sqrt {393}\right )^{{2}/{3}}}{4}+\left (190+6 \sqrt {393}\right )^{{1}/{3}}+7\right )}{3 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{\frac {\left (28+\left (190+6 \sqrt {393}\right )^{{2}/{3}}-8 \left (190+6 \sqrt {393}\right )^{{1}/{3}}\right ) t}{6 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{2}/{3}}-28\right ) t}{6 \left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{1}/{3}}}\right )+c_4 \,{\mathrm e}^{\frac {\left (28+\left (190+6 \sqrt {393}\right )^{{2}/{3}}-8 \left (190+6 \sqrt {393}\right )^{{1}/{3}}\right ) t}{6 \left (190+6 \sqrt {393}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{2}/{3}}-28\right ) t}{6 \left (190+6 \sqrt {3}\, \sqrt {131}\right )^{{1}/{3}}}\right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 95
ode=D[y[t],{t,4}]+5*D[y[t],{t,3}]+4*y[t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-4 \text {$\#$1}+4\&,1\right ]\right )+c_4 e^{-t}
\]
✓ Sympy. Time used: 0.461 (sec). Leaf size: 231
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(4*y(t) + 5*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = C_{1} e^{\frac {t \left (-8 + \frac {14 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {393} + 95}} + \sqrt [3]{2} \sqrt [3]{3 \sqrt {393} + 95}\right )}{6}} \sin {\left (\frac {\sqrt [3]{2} \sqrt {3} t \left (- \sqrt [3]{3 \sqrt {393} + 95} + \frac {14 \sqrt [3]{2}}{\sqrt [3]{3 \sqrt {393} + 95}}\right )}{6} \right )} + C_{2} e^{\frac {t \left (-8 + \frac {14 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {393} + 95}} + \sqrt [3]{2} \sqrt [3]{3 \sqrt {393} + 95}\right )}{6}} \cos {\left (\frac {\sqrt [3]{2} \sqrt {3} t \left (- \sqrt [3]{3 \sqrt {393} + 95} + \frac {14 \sqrt [3]{2}}{\sqrt [3]{3 \sqrt {393} + 95}}\right )}{6} \right )} + C_{3} e^{- t} + C_{4} e^{- \frac {t \left (4 + \frac {14 \cdot 2^{\frac {2}{3}}}{\sqrt [3]{3 \sqrt {393} + 95}} + \sqrt [3]{2} \sqrt [3]{3 \sqrt {393} + 95}\right )}{3}}
\]