72.20.1 problem 2
Internal
problem
ID
[14888]
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
:
2
Date
solved
:
Thursday, March 13, 2025 at 05:18:31 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+3 y&=5 \delta \left (t -2\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: 10.763 (sec). Leaf size: 21
ode:=diff(diff(y(t),t),t)+3*y(t) = 5*Dirac(t-2);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {5 \sqrt {3}\, \operatorname {Heaviside}\left (t -2\right ) \sin \left (\sqrt {3}\, \left (t -2\right )\right )}{3}
\]
✓ Mathematica. Time used: 0.088 (sec). Leaf size: 160
ode=D[y[t],{t,2}]+3*y[t]==DiracDelta[t-2];
ic={y[0]==2,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \cos \left (\sqrt {3} t\right ) \left (-\int _1^0-\frac {\delta (K[1]-2) \sin \left (2 \sqrt {3}\right )}{\sqrt {3}}dK[1]\right )+\cos \left (\sqrt {3} t\right ) \int _1^t-\frac {\delta (K[1]-2) \sin \left (2 \sqrt {3}\right )}{\sqrt {3}}dK[1]-\sin \left (\sqrt {3} t\right ) \int _1^0\frac {\cos \left (2 \sqrt {3}\right ) \delta (K[2]-2)}{\sqrt {3}}dK[2]+\sin \left (\sqrt {3} t\right ) \int _1^t\frac {\cos \left (2 \sqrt {3}\right ) \delta (K[2]-2)}{\sqrt {3}}dK[2]+2 \cos \left (\sqrt {3} t\right )
\]
✓ Sympy. Time used: 0.898 (sec). Leaf size: 114
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-5*Dirac(t - 2) + 3*y(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)
\[
y{\left (t \right )} = \left (- \frac {5 \sqrt {3} \int \operatorname {Dirac}{\left (t - 2 \right )} \sin {\left (\sqrt {3} t \right )}\, dt}{3} + \frac {5 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \sin {\left (\sqrt {3} t \right )}\, dt}{3}\right ) \cos {\left (\sqrt {3} t \right )} + \left (\frac {5 \sqrt {3} \int \operatorname {Dirac}{\left (t - 2 \right )} \cos {\left (\sqrt {3} t \right )}\, dt}{3} - \frac {5 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} \cos {\left (\sqrt {3} t \right )}\, dt}{3}\right ) \sin {\left (\sqrt {3} t \right )}
\]