73.22.13 problem 31.7 (f)

Internal problem ID [15569]
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 : 31.7 (f)
Date solved : Monday, March 31, 2025 at 01:41:24 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.071 (sec). Leaf size: 5

ode:=diff(diff(y(t),t),t)+y(t) = Dirac(t); 
ic:=y(0) = 0, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 37

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

\[ y(t)\to \sin (t) \left (-\int _1^0\delta (K[1])dK[1]\right )+\sin (t) \int _1^t\delta (K[1])dK[1]-\sin (t) \]

✓ Sympy. Time used: 0.714 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \int \operatorname {Dirac}{\left (t \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (\int \operatorname {Dirac}{\left (t \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \cos {\left (t \right )}\, dt - 1\right ) \sin {\left (t \right )} \]