72.20.4 problem 5
Internal
problem
ID
[14891]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
6.
Laplace
transform.
Section
6.4.
page
608
Problem
number
:
5
Date
solved
:
Thursday, March 13, 2025 at 05:18:37 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=\delta \left (t -1\right )-3 \delta \left (t -4\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: 12.681 (sec). Leaf size: 49
ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+3*y(t) = Dirac(t-1)-3*Dirac(t-4);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\sqrt {2}\, \left (-3 \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4} \sin \left (\sqrt {2}\, \left (t -4\right )\right )+\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t} \sin \left (\sqrt {2}\, \left (t -1\right )\right )\right )}{2}
\]
✓ Mathematica. Time used: 0.357 (sec). Leaf size: 217
ode=D[y[t],{t,2}]+2*D[y[t],t]+3*y[t]==DiracDelta[t-1]-3*DiracDelta[t-4];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -e^{-t} \left (\sin \left (\sqrt {2} t\right ) \int _1^0-\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (3 \delta (K[1]-4)-\delta (K[1]-1))}{\sqrt {2}}dK[1]-\sin \left (\sqrt {2} t\right ) \int _1^t-\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (3 \delta (K[1]-4)-\delta (K[1]-1))}{\sqrt {2}}dK[1]+\cos \left (\sqrt {2} t\right ) \int _1^0\frac {e^{K[2]} (3 \delta (K[2]-4)-\delta (K[2]-1)) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]-\cos \left (\sqrt {2} t\right ) \int _1^t\frac {e^{K[2]} (3 \delta (K[2]-4)-\delta (K[2]-1)) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]\right )
\]
✓ Sympy. Time used: 7.569 (sec). Leaf size: 192
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(3*Dirac(t - 4) - Dirac(t - 1) + 3*y(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)
\[
y{\left (t \right )} = \left (\left (- \frac {\sqrt {2} \int \left (- 3 \operatorname {Dirac}{\left (t - 4 \right )} + \operatorname {Dirac}{\left (t - 1 \right )}\right ) e^{t} \sin {\left (\sqrt {2} t \right )}\, dt}{2} - \frac {3 \sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \right )} e^{t} \sin {\left (\sqrt {2} t \right )}\, dt}{2} + \frac {\sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t} \sin {\left (\sqrt {2} t \right )}\, dt}{2}\right ) \cos {\left (\sqrt {2} t \right )} + \left (\frac {\sqrt {2} \int \left (- 3 \operatorname {Dirac}{\left (t - 4 \right )} + \operatorname {Dirac}{\left (t - 1 \right )}\right ) e^{t} \cos {\left (\sqrt {2} t \right )}\, dt}{2} + \frac {3 \sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \right )} e^{t} \cos {\left (\sqrt {2} t \right )}\, dt}{2} - \frac {\sqrt {2} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t} \cos {\left (\sqrt {2} t \right )}\, dt}{2}\right ) \sin {\left (\sqrt {2} t \right )}\right ) e^{- t}
\]