76.22.8 problem 21

Internal problem ID [17789]
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 : Tuesday, January 28, 2025 at 08:28:07 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*}

✓ Solution by Maple

Time used: 15.093 (sec). Leaf size: 93

dsolve([diff(y(t),t$4)+diff(y(t),t$2)+16*y(t)=g(t),y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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