11.1.1 problem 1
Internal
problem
ID
[1462]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
11th
ed.,
Boyce,
DiPrima,
Meade
Section
:
Chapter
4.1,
Higher
order
linear
differential
equations.
General
theory.
page
173
Problem
number
:
1
Date
solved
:
Tuesday, March 04, 2025 at 12:36:21 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y&=t \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 182
ode:=diff(diff(diff(diff(y(t),t),t),t),t)+4*diff(diff(diff(y(t),t),t),t)+3*y(t) = t;
dsolve(ode,y(t), singsol=all);
\[
y = \frac {t}{3}+{\mathrm e}^{-t} c_1 +c_2 \,{\mathrm e}^{\frac {t \left (\left (-2+\sqrt {2}\right ) \left (4+2 \sqrt {2}\right )^{{2}/{3}}-2 \left (4+2 \sqrt {2}\right )^{{1}/{3}}-2\right )}{2}}+c_3 \,{\mathrm e}^{-\frac {t \left (\left (-2+\sqrt {2}\right ) \left (4+2 \sqrt {2}\right )^{{2}/{3}}-2 \left (4+2 \sqrt {2}\right )^{{1}/{3}}+4\right )}{4}} \cos \left (\frac {\sqrt {3}\, t \left (4+2 \sqrt {2}\right )^{{1}/{3}} \left (2+\left (-2+\sqrt {2}\right ) \left (4+2 \sqrt {2}\right )^{{1}/{3}}\right )}{4}\right )+c_4 \,{\mathrm e}^{-\frac {t \left (\left (-2+\sqrt {2}\right ) \left (4+2 \sqrt {2}\right )^{{2}/{3}}-2 \left (4+2 \sqrt {2}\right )^{{1}/{3}}+4\right )}{4}} \sin \left (\frac {\sqrt {3}\, t \left (4+2 \sqrt {2}\right )^{{1}/{3}} \left (2+\left (-2+\sqrt {2}\right ) \left (4+2 \sqrt {2}\right )^{{1}/{3}}\right )}{4}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 100
ode=D[y[t],{t,4}]+4*D[ y[t],{t,3}]+3*y[t]==t;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+3\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+3\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-3 \text {$\#$1}+3\&,1\right ]\right )+\frac {t}{3}+c_4 e^{-t}
\]
✓ Sympy. Time used: 0.450 (sec). Leaf size: 202
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t + 3*y(t) + 4*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \frac {t}{3} + \left (C_{1} + C_{2} e^{\frac {\sqrt [3]{2} t \left (\frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {2} + 2}} + \sqrt [3]{\sqrt {2} + 2}\right )}{2}} \sin {\left (\frac {\sqrt [3]{2} \sqrt {3} t \left (- \sqrt [3]{\sqrt {2} + 2} + \frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {2} + 2}}\right )}{2} \right )} + C_{3} e^{\frac {\sqrt [3]{2} t \left (\frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {2} + 2}} + \sqrt [3]{\sqrt {2} + 2}\right )}{2}} \cos {\left (\frac {\sqrt [3]{2} \sqrt {3} t \left (- \sqrt [3]{\sqrt {2} + 2} + \frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {2} + 2}}\right )}{2} \right )} + C_{4} e^{- \sqrt [3]{2} t \left (\frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {2} + 2}} + \sqrt [3]{\sqrt {2} + 2}\right )}\right ) e^{- t}
\]