67.22.6 problem 31.6 (f)

Internal problem ID [16920]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.6 (f)
Date solved : Thursday, October 02, 2025 at 01:40:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method

✓ Maple. Time used: 0.289 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+y(t) = Dirac(t)+Dirac(t-Pi); 
dsolve(ode,y(t),method='laplace');
 

\[ y = y \left (0\right ) \cos \left (t \right )+\sin \left (t \right ) \left (\operatorname {Heaviside}\left (-t +\pi \right )+y^{\prime }\left (0\right )\right ) \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 113

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

\begin{align*} y(t)&\to \cos \left (\sqrt {2} t\right ) \int _1^t-\frac {(\delta (K[1])+\delta (K[1]-\pi )) \sin \left (\sqrt {2} K[1]\right )}{\sqrt {2}}dK[1]+\sin \left (\sqrt {2} t\right ) \int _1^t\frac {\cos \left (\sqrt {2} K[2]\right ) (\delta (K[2])+\delta (K[2]-\pi ))}{\sqrt {2}}dK[2]+c_1 \cos \left (\sqrt {2} t\right )+c_2 \sin \left (\sqrt {2} t\right ) \end{align*}

✓ Sympy. Time used: 0.934 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = \left (C_{1} - \int \left (\operatorname {Dirac}{\left (t \right )} + \operatorname {Dirac}{\left (t - \pi \right )}\right ) \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (C_{2} + \int \left (\operatorname {Dirac}{\left (t \right )} + \operatorname {Dirac}{\left (t - \pi \right )}\right ) \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} \]