85.71.3 problem 5 (b)

Internal problem ID [22935]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. C Exercises at page 284
Problem number : 5 (b)
Date solved : Thursday, October 02, 2025 at 09:16:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.258 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+y(t) = 3*Dirac(t-Pi); 
ic:=[y(0) = 6, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -3 \operatorname {Heaviside}\left (t -\pi \right ) \sin \left (t \right )+6 \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 20

ode=D[y[t],{t,2}]+y[t]==3*DiracDelta[t-Pi]; 
ic={y[0]==6,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 6 \cos (t)-3 \theta (t-\pi ) \sin (t) \end{align*}

✓ Sympy. Time used: 0.513 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - pi) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (3 \int \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (t \right )}\, dt - 3 \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} + \left (- 3 \int \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (t \right )}\, dt + 3 \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (t \right )}\, dt + 6\right ) \cos {\left (t \right )} \]