63.17.6 problem 9
Internal
problem
ID
[13126]
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
:
9
Date
solved
:
Wednesday, March 05, 2025 at 09:17:52 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\delta \left (t -1\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: 10.645 (sec). Leaf size: 28
ode:=diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = Dirac(t-1);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {2 \operatorname {Heaviside}\left (t -1\right ) \sqrt {3}\, {\mathrm e}^{\frac {1}{2}-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, \left (t -1\right )}{2}\right )}{3}
\]
✓ Mathematica. Time used: 0.076 (sec). Leaf size: 197
ode=D[y[t],{t,2}]+D[y[t],t]+y[t]==DiracDelta[t-1];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -e^{-t/2} \left (\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^02 \sqrt {\frac {e}{3}} \cos \left (\frac {\sqrt {3}}{2}\right ) \delta (K[1]-1)dK[1]-\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t2 \sqrt {\frac {e}{3}} \cos \left (\frac {\sqrt {3}}{2}\right ) \delta (K[1]-1)dK[1]+\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^0-2 \sqrt {\frac {e}{3}} \delta (K[2]-1) \sin \left (\frac {\sqrt {3}}{2}\right )dK[2]-\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-2 \sqrt {\frac {e}{3}} \delta (K[2]-1) \sin \left (\frac {\sqrt {3}}{2}\right )dK[2]\right )
\]
✓ Sympy. Time used: 2.123 (sec). Leaf size: 150
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-Dirac(t - 1) + y(t) + 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)
\[
y{\left (t \right )} = \left (\left (- \frac {2 \sqrt {3} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} + \frac {2 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3}\right ) \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {2 \sqrt {3} \int \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} - \frac {2 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3}\right ) \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}}
\]