76.21.10 problem 10
Internal
problem
ID
[17695]
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
:
10
Date
solved
:
Thursday, March 13, 2025 at 10:47:21 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 2 y^{\prime \prime }+y^{\prime }+6 y&=\delta \left (t -\frac {\pi }{6}\right ) \sin \left (t \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: 14.643 (sec). Leaf size: 34
ode:=2*diff(diff(y(t),t),t)+diff(y(t),t)+6*y(t) = Dirac(t-1/6*Pi)*sin(t);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = -\frac {\sqrt {47}\, {\mathrm e}^{\frac {\pi }{24}-\frac {t}{4}} \operatorname {Heaviside}\left (t -\frac {\pi }{6}\right ) \sin \left (\frac {\sqrt {47}\, \left (-6 t +\pi \right )}{24}\right )}{47}
\]
✓ Mathematica. Time used: 0.091 (sec). Leaf size: 46
ode=2*D[y[t],{t,2}]+D[y[t],t]+6*y[t]==DiracDelta[t-Pi/6]*Sin[t];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -\frac {e^{\frac {1}{24} (\pi -6 t)} \theta (6 t-\pi ) \sin \left (\frac {1}{24} \sqrt {47} (\pi -6 t)\right )}{\sqrt {47}}
\]
✓ Sympy. Time used: 15.438 (sec). Leaf size: 170
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-Dirac(t - pi/6)*sin(t) + 6*y(t) + Derivative(y(t), t) + 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)
\[
y{\left (t \right )} = \left (\left (- \frac {2 \sqrt {47} \int \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \sin {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47} + \frac {2 \sqrt {47} \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \sin {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47}\right ) \cos {\left (\frac {\sqrt {47} t}{4} \right )} + \left (\frac {2 \sqrt {47} \int \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \cos {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47} - \frac {2 \sqrt {47} \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{6} \right )} e^{\frac {t}{4}} \sin {\left (t \right )} \cos {\left (\frac {\sqrt {47} t}{4} \right )}\, dt}{47}\right ) \sin {\left (\frac {\sqrt {47} t}{4} \right )}\right ) e^{- \frac {t}{4}}
\]