76.22.7 problem 20

Internal problem ID [17788]
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 : Tuesday, January 28, 2025 at 11:03:05 AM
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*}

✓ Solution by Maple

Time used: 18.073 (sec). Leaf size: 62

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

\[ y = -\frac {\left (\int _{0}^{t}\sin \left (2 t -2 \textit {\_U1} \right ) g \left (\textit {\_U1} \right )d \textit {\_U1} \right )}{16}-\frac {\left (\int _{0}^{t}{\mathrm e}^{-2 t +2 \textit {\_U1}} g \left (\textit {\_U1} \right )d \textit {\_U1} \right )}{32}+\frac {\left (\int _{0}^{t}{\mathrm e}^{2 t -2 \textit {\_U1}} g \left (\textit {\_U1} \right )d \textit {\_U1} \right )}{32} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 210

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

\[ y(t)\to -e^{2 t} \int _1^0\frac {1}{32} e^{-2 K[1]} g(K[1])dK[1]+e^{2 t} \int _1^t\frac {1}{32} e^{-2 K[1]} g(K[1])dK[1]-e^{-2 t} \int _1^0-\frac {1}{32} e^{2 K[3]} g(K[3])dK[3]+e^{-2 t} \int _1^t-\frac {1}{32} e^{2 K[3]} g(K[3])dK[3]-\sin (2 t) \int _1^0-\frac {1}{16} \cos (2 K[4]) g(K[4])dK[4]+\sin (2 t) \int _1^t-\frac {1}{16} \cos (2 K[4]) g(K[4])dK[4]-\cos (2 t) \int _1^0\frac {1}{16} g(K[2]) \sin (2 K[2])dK[2]+\cos (2 t) \int _1^t\frac {1}{16} g(K[2]) \sin (2 K[2])dK[2] \]