46.8.5 problem 7

Internal problem ID [7376]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number : 7
Date solved : Monday, January 27, 2025 at 02:51:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+24 y^{\prime }+37 y&=17 \,{\mathrm e}^{-t}+\delta \left (t -\frac {1}{2}\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.493 (sec). Leaf size: 37

dsolve([4*diff(y(t),t$2)+24*diff(y(t),t)+37*y(t)=17*exp(-t)+Dirac(t-1/2),y(0) = 1, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (t -\frac {1}{2}\right ) {\mathrm e}^{-3 t +\frac {3}{2}} \sin \left (-\frac {1}{4}+\frac {t}{2}\right )}{2}+4 \,{\mathrm e}^{-3 t} \sin \left (\frac {t}{2}\right )+{\mathrm e}^{-t} \]

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 63

DSolve[{4*D[y[t],{t,2}]+24*D[y[t],t]+27*y[t]==17*Exp[-t]+DiracDelta[t-1/2],{y[0]==1,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{84} e^{-9 t/2} \left (7 e^{3/4} \left (e^{3 t}-e^{3/2}\right ) \theta (2 t-1)+12 \left (-7 e^{3 t}+17 e^{7 t/2}-3\right )\right ) \]