76.22.7 problem 20
Internal
problem
ID
[17717]
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
5.8
(Convolution
Integrals
and
Their
Applications).
Problems
at
page
359
Problem
number
:
20
Date
solved
:
Monday, March 31, 2025 at 04:25:52 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }-16 y&=g \left (t \right ) \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 )&=0 \end{align*}
✓ Maple. Time used: 0.370 (sec). Leaf size: 62
ode:=diff(diff(diff(diff(y(t),t),t),t),t)-16*y(t) = g(t);
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\int _{0}^{t}g \left (\textit {\_U1} \right ) {\mathrm e}^{2 t -2 \textit {\_U1}}d \textit {\_U1}}{32}-\frac {\int _{0}^{t}g \left (\textit {\_U1} \right ) {\mathrm e}^{-2 t +2 \textit {\_U1}}d \textit {\_U1}}{32}-\frac {\int _{0}^{t}g \left (\textit {\_U1} \right ) \sin \left (2 t -2 \textit {\_U1} \right )d \textit {\_U1}}{16}
\]
✓ Mathematica. Time used: 0.039 (sec). Leaf size: 210
ode=D[y[t],{t,4}]-16*y[t]==g[t];
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -e^{2 t} \int _1^0\frac {1}{32} e^{-2 K[1]} g(K[1])dK[1]+e^{2 t} \int _1^t\frac {1}{32} e^{-2 K[1]} g(K[1])dK[1]-e^{-2 t} \int _1^0-\frac {1}{32} e^{2 K[3]} g(K[3])dK[3]+e^{-2 t} \int _1^t-\frac {1}{32} e^{2 K[3]} g(K[3])dK[3]-\sin (2 t) \int _1^0-\frac {1}{16} \cos (2 K[4]) g(K[4])dK[4]+\sin (2 t) \int _1^t-\frac {1}{16} \cos (2 K[4]) g(K[4])dK[4]-\cos (2 t) \int _1^0\frac {1}{16} g(K[2]) \sin (2 K[2])dK[2]+\cos (2 t) \int _1^t\frac {1}{16} g(K[2]) \sin (2 K[2])dK[2]
\]
✓ Sympy. Time used: 2.069 (sec). Leaf size: 117
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-g(t) - 16*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): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\frac {\int g{\left (t \right )} e^{- 2 t}\, dt}{32} - \frac {\int \limits ^{0} g{\left (t \right )} e^{- 2 t}\, dt}{32}\right ) e^{2 t} + \left (- \frac {\int g{\left (t \right )} e^{2 t}\, dt}{32} + \frac {\int \limits ^{0} g{\left (t \right )} e^{2 t}\, dt}{32}\right ) e^{- 2 t} + \left (\frac {\int g{\left (t \right )} \sin {\left (2 t \right )}\, dt}{16} - \frac {\int \limits ^{0} g{\left (t \right )} \sin {\left (2 t \right )}\, dt}{16}\right ) \cos {\left (2 t \right )} + \left (- \frac {\int g{\left (t \right )} \cos {\left (2 t \right )}\, dt}{16} + \frac {\int \limits ^{0} g{\left (t \right )} \cos {\left (2 t \right )}\, dt}{16}\right ) \sin {\left (2 t \right )}
\]