76.18.10 problem 21
Internal
problem
ID
[17650]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
5.
The
Laplace
transform.
Section
5.2
(Properties
of
the
Laplace
transform).
Problems
at
page
309
Problem
number
:
21
Date
solved
:
Monday, March 31, 2025 at 04:23:37 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }-6 y&=t \,{\mathrm e}^{-t} \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=9 \end{align*}
✓ Maple. Time used: 0.226 (sec). Leaf size: 84
ode:=diff(diff(diff(diff(y(t),t),t),t),t)-6*y(t) = t*exp(-t);
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 9;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\left (-12-7 \sqrt {6}\right ) \cosh \left (6^{{1}/{4}} t \right )}{150}+\frac {\left (9 \,6^{{3}/{4}}+244 \,6^{{1}/{4}}\right ) \sinh \left (6^{{1}/{4}} t \right )}{300}+\frac {\cos \left (6^{{1}/{4}} t \right ) \left (-12+7 \sqrt {6}\right )}{150}+\frac {\left (9 \,6^{{3}/{4}}-244 \,6^{{1}/{4}}\right ) \sin \left (6^{{1}/{4}} t \right )}{300}+\frac {\left (-5 t +4\right ) {\mathrm e}^{-t}}{25}
\]
✓ Mathematica. Time used: 0.012 (sec). Leaf size: 190
ode=D[y[t],{t,4}]-6*y[t]==t*Exp[-t];
ic={y[0]==0,Derivative[1][y][0] == 0,Derivative[2][y][0] == 0,Derivative[3][y][0] == 9};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{600} \left (-120 e^{-t} t+96 e^{-t}-9\ 6^{3/4} e^{-\sqrt [4]{6} t}-14 \sqrt {6} e^{-\sqrt [4]{6} t}-244 \sqrt [4]{6} e^{-\sqrt [4]{6} t}-24 e^{-\sqrt [4]{6} t}+9\ 6^{3/4} e^{\sqrt [4]{6} t}-14 \sqrt {6} e^{\sqrt [4]{6} t}+244 \sqrt [4]{6} e^{\sqrt [4]{6} t}-24 e^{\sqrt [4]{6} t}+2 \sqrt [4]{6} \left (9 \sqrt {6}-244\right ) \sin \left (\sqrt [4]{6} t\right )+4 \left (7 \sqrt {6}-12\right ) \cos \left (\sqrt [4]{6} t\right )\right )
\]
✓ Sympy. Time used: 0.280 (sec). Leaf size: 138
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t*exp(-t) - 6*y(t) + Derivative(y(t), (t, 4)),0)
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 9}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = - \frac {t e^{- t}}{5} + \left (- \frac {7 \sqrt {6}}{300} - \frac {1}{25} + \frac {3 \cdot 6^{\frac {3}{4}}}{200} + \frac {61 \sqrt [4]{6}}{150}\right ) e^{\sqrt [4]{6} t} + \left (- \frac {61 \sqrt [4]{6}}{75} + \frac {3 \cdot 6^{\frac {3}{4}}}{100}\right ) \sin {\left (\sqrt [4]{6} t \right )} + \left (- \frac {2}{25} + \frac {7 \sqrt {6}}{150}\right ) \cos {\left (\sqrt [4]{6} t \right )} + \left (- \frac {61 \sqrt [4]{6}}{150} - \frac {3 \cdot 6^{\frac {3}{4}}}{200} - \frac {7 \sqrt {6}}{300} - \frac {1}{25}\right ) e^{- \sqrt [4]{6} t} + \frac {4 e^{- t}}{25}
\]