76.21.9 problem 9
Internal
problem
ID
[17694]
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, March 13, 2025 at 10:47:19 AM
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: 13.088 (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,ic],y(t),method='laplace');
\[
y = \left (1-\sin \left (t \right )\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+3 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\frac {3 \pi }{2}\right )+\left (\cos \left (t \right )-1\right ) \operatorname {Heaviside}\left (t -2 \pi \right )
\]
✓ Mathematica. Time used: 0.331 (sec). Leaf size: 75
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]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 3 \cos (t) \theta (2 t-3 \pi ) & 2 t\leq \pi \\ 3 \cos (t) \theta (2 t-3 \pi )-\sin (t)+1 & \frac {\pi }{2}<t\leq 2 \pi \\ 3 \theta (2 t-3 \pi ) \cos (t)+\cos (t)-\sin (t) & \text {True} \\ \end {array} \\ \end {array}
\]
✓ Sympy. Time used: 2.301 (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)
\[
y{\left (t \right )} = \left (- \int \left (3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} - \theta \left (t - 2 \pi \right ) + \theta \left (t - \frac {\pi }{2}\right )\right ) \sin {\left (t \right )}\, dt + \int \limits ^{0} 3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \sin {\left (t \right )}\, dt + \int \limits ^{0} \left (- \sin {\left (t \right )} \theta \left (t - 2 \pi \right )\right )\, dt + \int \limits ^{0} \sin {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right )\, dt\right ) \cos {\left (t \right )} + \left (\int \left (3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} - \theta \left (t - 2 \pi \right ) + \theta \left (t - \frac {\pi }{2}\right )\right ) \cos {\left (t \right )}\, dt - \int \limits ^{0} 3 \operatorname {Dirac}{\left (t - \frac {3 \pi }{2} \right )} \cos {\left (t \right )}\, dt - \int \limits ^{0} \left (- \cos {\left (t \right )} \theta \left (t - 2 \pi \right )\right )\, dt - \int \limits ^{0} \cos {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right )\, dt\right ) \sin {\left (t \right )}
\]