90.12.3 problem 3
Internal
problem
ID
[25205]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
3.
Second
Order
Constant
Coefficient
Linear
Differential
Equations.
Exercises
at
page
213
Problem
number
:
3
Date
solved
:
Thursday, October 02, 2025 at 11:58:04 PM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime }+y^{\prime }+4 y&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 149
ode:=diff(diff(diff(y(t),t),t),t)+diff(y(t),t)+4*y(t) = 0;
dsolve(ode,y(t), singsol=all);
\[
y = \left (c_2 \,{\mathrm e}^{\frac {\left (\left (54+3 \sqrt {327}\right )^{{2}/{3}}-3\right ) t}{2 \left (54+3 \sqrt {327}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (54+3 \sqrt {3}\, \sqrt {109}\right )^{{2}/{3}}+3\right ) t}{6 \left (54+3 \sqrt {3}\, \sqrt {109}\right )^{{1}/{3}}}\right )+c_3 \,{\mathrm e}^{\frac {\left (\left (54+3 \sqrt {327}\right )^{{2}/{3}}-3\right ) t}{2 \left (54+3 \sqrt {327}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (54+3 \sqrt {3}\, \sqrt {109}\right )^{{2}/{3}}+3\right ) t}{6 \left (54+3 \sqrt {3}\, \sqrt {109}\right )^{{1}/{3}}}\right )+c_1 \right ) {\mathrm e}^{-\frac {\left (\left (54+3 \sqrt {327}\right )^{{2}/{3}}-3\right ) t}{3 \left (54+3 \sqrt {327}\right )^{{1}/{3}}}}
\]
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 63
ode=D[y[t],{t,3}]+D[y[t],{t,1}]+4*y[t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+4\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+4\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+4\&,1\right ]\right ) \end{align*}
✓ Sympy. Time used: 0.274 (sec). Leaf size: 189
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(4*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 3)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = C_{1} e^{- \frac {\sqrt [3]{3} t \left (- \sqrt [3]{18 + \sqrt {327}} + \frac {\sqrt [3]{3}}{\sqrt [3]{18 + \sqrt {327}}}\right )}{6}} \sin {\left (\frac {\sqrt [6]{3} t \left (\frac {3}{\sqrt [3]{18 + \sqrt {327}}} + 3^{\frac {2}{3}} \sqrt [3]{18 + \sqrt {327}}\right )}{6} \right )} + C_{2} e^{- \frac {\sqrt [3]{3} t \left (- \sqrt [3]{18 + \sqrt {327}} + \frac {\sqrt [3]{3}}{\sqrt [3]{18 + \sqrt {327}}}\right )}{6}} \cos {\left (\frac {\sqrt [6]{3} t \left (\frac {3}{\sqrt [3]{18 + \sqrt {327}}} + 3^{\frac {2}{3}} \sqrt [3]{18 + \sqrt {327}}\right )}{6} \right )} + C_{3} e^{\frac {\sqrt [3]{3} t \left (- \sqrt [3]{18 + \sqrt {327}} + \frac {\sqrt [3]{3}}{\sqrt [3]{18 + \sqrt {327}}}\right )}{3}}
\]