90.29.3 problem 12

Internal problem ID [25435]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 449
Problem number : 12
Date solved : Friday, October 03, 2025 at 12:01:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\delta \left (t -1\right )-\delta \left (t -2\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: 0.127 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)-y(t) = Dirac(t-1)-Dirac(t-2); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \sinh \left (t -2\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 50

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-t-2} \left (\left (e^4-e^{2 t}\right ) \theta (t-2)+e \left (e^{2 t}-e^2\right ) \theta (t-1)\right ) \end{align*}

✓ Sympy. Time used: 0.980 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Dirac(t - 2) - Dirac(t - 1) - y(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)
 

\[ y{\left (t \right )} = \left (- \frac {\int \operatorname {Dirac}{\left (t - 2 \right )} e^{- t}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{- t}\, dt}{2} + \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{2}\right ) e^{t} + \left (\frac {\int \operatorname {Dirac}{\left (t - 2 \right )} e^{t}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{t}\, dt}{2} - \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt}{2}\right ) e^{- t} \]