46.8.7 problem 9

Internal problem ID [7378]
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 : 9
Date solved : Monday, January 27, 2025 at 02:51:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\left (1-\operatorname {Heaviside}\left (t -10\right )\right ) {\mathrm e}^{t}-{\mathrm e}^{10} \delta \left (t -10\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.547 (sec). Leaf size: 59

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=(1-Heaviside(t-10))*exp(t)-exp(10)*Dirac(t-10),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-2 t} \left (\left (-{\mathrm e}^{3 t}+\left (\left (\sin \left (10\right )-7 \cos \left (10\right )\right ) \sin \left (t \right )+\left (7 \sin \left (10\right )+\cos \left (10\right )\right ) \cos \left (t \right )\right ) {\mathrm e}^{30}\right ) \operatorname {Heaviside}\left (t -10\right )-\cos \left (t \right )+7 \sin \left (t \right )+{\mathrm e}^{3 t}\right )}{10} \]

✓ Solution by Mathematica

Time used: 0.532 (sec). Leaf size: 94

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==(1-UnitStep[t-10])*Exp[t]-Exp[10]*DiracDelta[t-10],{y[0]==0,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{10} e^{-2 t} \left (10 e^{30} \theta (t-10) \sin (10-t)+\theta (10-t) \left (e^{3 t}+3 e^{30} \sin (10-t)-e^{30} \cos (10-t)\right )-3 e^{30} \sin (10-t)+7 \sin (t)+e^{30} \cos (10-t)-\cos (t)\right ) \]