70.21.9 problem 9

Internal problem ID [19059]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 9
Date solved : Thursday, October 02, 2025 at 03:37:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+3 \delta \left (t -\frac {3 \pi }{2}\right )-\operatorname {Heaviside}\left (t -2 \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: 0.415 (sec). Leaf size: 39

ode:=diff(diff(y(t),t),t)+y(t) = Heaviside(t-1/2*Pi)+3*Dirac(t-3/2*Pi)-Heaviside(t-2*Pi); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (-\sin \left (t \right )+1\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+3 \operatorname {Heaviside}\left (t -\frac {3 \pi }{2}\right ) \cos \left (t \right )+\operatorname {Heaviside}\left (t -2 \pi \right ) \left (\cos \left (t \right )-1\right ) \]

✓ Mathematica. Time used: 1.573 (sec). Leaf size: 313

ode=D[y[t],{t,2}]+y[t]==UnitStep[t-Pi/2]+3*DiracDelta[t-3*Pi/2]-UnitStep[t-2*Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \theta (2 \pi -t) \left (\cos (t) \left (-\int _1^t6 \delta (3 \pi -2 K[3])dK[3]\right )+\cos (t) \int _1^t-((6 \delta (2 K[2]-3 \pi )+1) \sin (K[2]))dK[2]+\sin (t) \int _1^t\cos (K[3]) (6 \delta (2 K[3]-3 \pi )+1)dK[3]+\sin (1) \sin (t)+\cos (1) \cos (t)-\cos (t)\right )+\theta \left (\frac {1}{2} (\pi -2 t)\right ) \left (\theta (2 \pi -t) \left (\cos (t) \int _1^06 \delta (3 \pi -2 K[1])dK[1]-\cos (t) \int _1^t-((6 \delta (2 K[2]-3 \pi )+1) \sin (K[2]))dK[2]-\sin (t) \int _1^t\cos (K[3]) (6 \delta (2 K[3]-3 \pi )+1)dK[3]-\sin (1) \sin (t)+\sin (t)-\cos (1) \cos (t)\right )-\cos (t) \int _1^06 \delta (3 \pi -2 K[1])dK[1]+\cos (t) \int _1^t6 \delta (3 \pi -2 K[1])dK[1]\right )+\cos (t) \left (-\int _1^06 \delta (3 \pi -2 K[1])dK[1]\right )+\cos (t) \int _1^t6 \delta (3 \pi -2 K[3])dK[3]-\sin (t)+\cos (t) \end{align*}

✓ Sympy. Time used: 1.569 (sec). Leaf size: 162

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - 3*pi/2) + y(t) + Heaviside(t - 2*pi) - Heaviside(t - pi/2) + 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)
 

\[ y{\left (t \right )} = \left (- \int \left (3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} - \theta \left (t - 2 \pi \right ) + \theta \left (t - \frac {\pi }{2}\right )\right ) \sin {\left (t \right )}\, dt + \int \limits ^{0} 3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} \left (- \sin {\left (t \right )} \theta \left (t - 2 \pi \right )\right )\, dt + \int \limits ^{0} \sin {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right )\, dt\right ) \cos {\left (t \right )} + \left (\int \left (3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} - \theta \left (t - 2 \pi \right ) + \theta \left (t - \frac {\pi }{2}\right )\right ) \cos {\left (t \right )}\, dt - \int \limits ^{0} 3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} \left (- \cos {\left (t \right )} \theta \left (t - 2 \pi \right )\right )\, dt - \int \limits ^{0} \cos {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right )\, dt\right ) \sin {\left (t \right )} \]