76.22.8 problem 21

Internal problem ID [17718]
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 : 21
Date solved : Monday, March 31, 2025 at 04:25:53 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }+16 y&=g \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ 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: 0.234 (sec). Leaf size: 93

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+diff(diff(y(t),t),t)+16*y(t) = g(t); 
ic:=y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+16\right )}{\sum }\left (\underline {\hspace {1.25 ex}}\alpha ^{2}+32\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{63}+\frac {\int _{0}^{t}g \left (\textit {\_U1} \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+16\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-31\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha \left (t -\textit {\_U1} \right )}\right )d \textit {\_U1}}{2016}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+16\right )}{\sum }\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{63} \]

✓ Mathematica. Time used: 1.158 (sec). Leaf size: 40549

ode=D[y[t],{t,4}]+D[y[t],t]+16*y[t]==g[t]; 
ic={y[0]==2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Too large to display

✓ Sympy. Time used: 36.507 (sec). Leaf size: 835

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-g(t) + 16*y(t) + Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 2, 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)
 

\[ \text {Solution too large to show} \]