46.8.4 problem 4

Internal problem ID [9654]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 06:21:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.123 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 53

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

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

✓ Sympy. Time used: 0.653 (sec). Leaf size: 73

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

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