14.19.2 problem 5

Internal problem ID [2695]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.12, Dirac delta function. Excercises page 250
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:14:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sin \left (t \right )+\delta \left (t -\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: 0.285 (sec). Leaf size: 24

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

\[ y = \frac {\sin \left (t \right ) \left (3 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )-\cos \left (t \right )+1\right )}{3} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 27

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

\[ y(t)\to \frac {1}{3} \sin (t) (3 \theta (t-\pi ) \cos (t)-\cos (t)+1) \]

✓ Sympy. Time used: 2.462 (sec). Leaf size: 97

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi) + 4*y(t) - sin(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)
 

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