40.8.1 problem 3

Internal problem ID [8658]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:40:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=8 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.203 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = Dirac(t-Pi); 
ic:=[y(0) = 8, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\operatorname {Heaviside}\left (t -\pi \right ) \sin \left (2 t \right )}{2}+8 \cos \left (2 t \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 59

ode=D[y[t],{t,2}]+4*y[t]==DiracDelta[t-Pi]; 
ic={y[0]==8,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sin (2 t) \int _1^0\frac {1}{2} \delta (\pi -K[1])dK[1]+\sin (2 t) \int _1^t\frac {1}{2} \delta (\pi -K[1])dK[1]+8 \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.744 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 8, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )}\, dt}{2} + 8\right ) \cos {\left (2 t \right )} \]