40.8.2 problem 4

Internal problem ID [8659]
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 : 4
Date solved : Tuesday, September 30, 2025 at 05:40:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=4 \delta \left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.206 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+16*y(t) = 4*Dirac(t-3*Pi); 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (4 t \right )+2 \cos \left (4 t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 51

ode=D[y[t],{t,2}]+16*y[t]==4*DiracDelta[t-3*Pi]; 
ic={y[0]==2,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sin (4 t) \int _1^0\delta (K[1]-3 \pi )dK[1]+\sin (4 t) \int _1^t\delta (K[1]-3 \pi )dK[1]+2 \cos (4 t) \end{align*}

✓ Sympy. Time used: 0.592 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*Dirac(t - 3*pi) + 16*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\int \operatorname {Dirac}{\left (t - 3 \pi \right )} \cos {\left (4 t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \pi \right )} \cos {\left (4 t \right )}\, dt\right ) \sin {\left (4 t \right )} + \left (- \int \operatorname {Dirac}{\left (t - 3 \pi \right )} \sin {\left (4 t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \pi \right )} \sin {\left (4 t \right )}\, dt + 2\right ) \cos {\left (4 t \right )} \]