76.22.3 problem 16

Internal problem ID [17784]
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 : 16
Date solved : Tuesday, January 28, 2025 at 11:03:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+17 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 \end{align*}

✓ Solution by Maple

Time used: 35.071 (sec). Leaf size: 31

dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+17*y(t)=g(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\left (\int _{0}^{t}\sin \left (-2 t +2 \textit {\_U1} \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\textit {\_U1}}{2}} g \left (\textit {\_U1} \right )d \textit {\_U1} \right )}{8} \]

✓ Solution by Mathematica

Time used: 0.125 (sec). Leaf size: 141

DSolve[{4*D[y[t],{t,2}]+4*D[y[t],t]+17*y[t]==g[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t/2} \left (-\sin (2 t) \int _1^0\frac {1}{8} e^{\frac {K[1]}{2}} \cos (2 K[1]) g(K[1])dK[1]+\sin (2 t) \int _1^t\frac {1}{8} e^{\frac {K[1]}{2}} \cos (2 K[1]) g(K[1])dK[1]+\cos (2 t) \left (\int _1^t-\frac {1}{8} e^{\frac {K[2]}{2}} g(K[2]) \sin (2 K[2])dK[2]-\int _1^0-\frac {1}{8} e^{\frac {K[2]}{2}} g(K[2]) \sin (2 K[2])dK[2]\right )\right ) \]