63.17.4 problem 6

Internal problem ID [13124]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page 173
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 09:17:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\delta \left (t -2\right )-\delta \left (t -5\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 10.032 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)+4*x(t) = Dirac(t-2)-Dirac(t-5); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -5\right ) \sin \left (-10+2 t \right )}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) \sin \left (2 t -4\right )}{2} \]

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 140

ode=D[x[t],{t,2}]+4*x[t]==DiracDelta[t-2]-DiracDelta[t-5]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\sin (2 t) \int _1^0-\frac {1}{2} \cos (2 K[2]) (\delta (K[2]-5)-\delta (K[2]-2))dK[2]+\sin (2 t) \int _1^t-\frac {1}{2} \cos (2 K[2]) (\delta (K[2]-5)-\delta (K[2]-2))dK[2]-\cos (2 t) \int _1^0\cos (K[1]) (\delta (K[1]-5)-\delta (K[1]-2)) \sin (K[1])dK[1]+\cos (2 t) \int _1^t\cos (K[1]) (\delta (K[1]-5)-\delta (K[1]-2)) \sin (K[1])dK[1] \]

✓ Sympy. Time used: 2.484 (sec). Leaf size: 107

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(Dirac(t - 5) - Dirac(t - 2) + 4*x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {\int \left (- \operatorname {Dirac}{\left (t - 5 \right )} + \operatorname {Dirac}{\left (t - 2 \right )}\right ) \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \left (- \operatorname {Dirac}{\left (t - 5 \right )} \sin {\left (2 t \right )}\right )\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \sin {\left (2 t \right )}\, dt}{2}\right ) \cos {\left (2 t \right )} + \left (\frac {\int \left (- \operatorname {Dirac}{\left (t - 5 \right )} + \operatorname {Dirac}{\left (t - 2 \right )}\right ) \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \left (- \operatorname {Dirac}{\left (t - 5 \right )} \cos {\left (2 t \right )}\right )\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} \]