76.21.5 problem 5
Internal
problem
ID
[17698]
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
:
5
Date
solved
:
Monday, March 31, 2025 at 04:25:11 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=\sin \left (t \right )+\delta \left (t -3 \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: 0.390 (sec). Leaf size: 54
ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+3*y(t) = sin(t)+Dirac(t-3*Pi);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {{\mathrm e}^{3 \pi -t} \sqrt {2}\, \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (\sqrt {2}\, \left (t -3 \pi \right )\right )}{2}+\frac {{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )}{4}-\frac {\cos \left (t \right )}{4}+\frac {\sin \left (t \right )}{4}
\]
✓ Mathematica. Time used: 1.598 (sec). Leaf size: 66
ode=D[y[t],{t,2}]+2*D[y[t],t]+3*y[t]==Sin[t]+DiracDelta[t-3*Pi];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{4} \left (-2 \sqrt {2} e^{3 \pi -t} \theta (t-3 \pi ) \sin \left (\sqrt {2} (3 \pi -t)\right )+\sin (t)-\cos (t)+e^{-t} \cos \left (\sqrt {2} t\right )\right )
\]
✓ Sympy. Time used: 157.226 (sec). Leaf size: 185
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-Dirac(t - 3*pi) + 3*y(t) - sin(t) + 2*Derivative(y(t), 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)
\[
y{\left (t \right )} = \left (\left (- \frac {\sqrt {2} \int \left (\operatorname {Dirac}{\left (t - 3 \pi \right )} + \sin {\left (t \right )}\right ) e^{t} \sin {\left (\sqrt {2} t \right )}\, dt}{2} + \frac {\sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \pi \right )} e^{t} \sin {\left (\sqrt {2} t \right )}\, dt}{2} + \frac {\sqrt {2} \int \limits ^{0} e^{t} \sin {\left (t \right )} \sin {\left (\sqrt {2} t \right )}\, dt}{2}\right ) \cos {\left (\sqrt {2} t \right )} + \left (\frac {\sqrt {2} \int \left (\operatorname {Dirac}{\left (t - 3 \pi \right )} + \sin {\left (t \right )}\right ) e^{t} \cos {\left (\sqrt {2} t \right )}\, dt}{2} - \frac {\sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \pi \right )} e^{t} \cos {\left (\sqrt {2} t \right )}\, dt}{2} - \frac {\sqrt {2} \int \limits ^{0} e^{t} \sin {\left (t \right )} \cos {\left (\sqrt {2} t \right )}\, dt}{2}\right ) \sin {\left (\sqrt {2} t \right )}\right ) e^{- t}
\]