67.4.30 problem Problem 5(a)
Internal
problem
ID
[14003]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
5.6
Laplace
transform.
Nonhomogeneous
equations.
Problems
page
368
Problem
number
:
Problem
5(a)
Date
solved
:
Monday, March 31, 2025 at 08:21:31 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+4 \pi ^{2} y&=3 \delta \left (t -\frac {1}{3}\right )-\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: 0.406 (sec). Leaf size: 44
ode:=diff(diff(y(t),t),t)+4*Pi^2*y(t) = 3*Dirac(t-1/3)-Dirac(t-1);
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\left (-3 \sqrt {3}\, \cos \left (2 \pi t \right )-3 \sin \left (2 \pi t \right )\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )-2 \sin \left (2 \pi t \right ) \operatorname {Heaviside}\left (t -1\right )}{4 \pi }
\]
✓ Mathematica. Time used: 0.12 (sec). Leaf size: 172
ode=D[y[t],{t,2}]+(2*Pi)^2*y[t]==3*DiracDelta[t-1/3]-DiracDelta[t-1];
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -\sin (2 \pi t) \int _1^0-\frac {\cos (2 \pi K[2]) (\delta (K[2]-1)-9 \delta (3 K[2]-1))}{2 \pi }dK[2]+\sin (2 \pi t) \int _1^t-\frac {\cos (2 \pi K[2]) (\delta (K[2]-1)-9 \delta (3 K[2]-1))}{2 \pi }dK[2]-\cos (2 \pi t) \int _1^0\frac {(\delta (K[1]-1)-9 \delta (3 K[1]-1)) \sin (2 \pi K[1])}{2 \pi }dK[1]+\cos (2 \pi t) \int _1^t\frac {(\delta (K[1]-1)-9 \delta (3 K[1]-1)) \sin (2 \pi K[1])}{2 \pi }dK[1]
\]
✓ Sympy. Time used: 1.989 (sec). Leaf size: 139
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(Dirac(t - 1) - 3*Dirac(t - 1/3) + 4*pi**2*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)
\[
y{\left (t \right )} = \left (\frac {- \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (2 \pi t \right )}\, dt - \int \limits ^{0} \left (- 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )} \sin {\left (2 \pi t \right )}\right )\, dt}{2 \pi } + \frac {\int \left (\operatorname {Dirac}{\left (t - 1 \right )} - 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )}\right ) \sin {\left (2 \pi t \right )}\, dt}{2 \pi }\right ) \cos {\left (2 \pi t \right )} + \left (\frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (2 \pi t \right )}\, dt + \int \limits ^{0} \left (- 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )} \cos {\left (2 \pi t \right )}\right )\, dt}{2 \pi } - \frac {\int \left (\operatorname {Dirac}{\left (t - 1 \right )} - 3 \operatorname {Dirac}{\left (t - \frac {1}{3} \right )}\right ) \cos {\left (2 \pi t \right )}\, dt}{2 \pi }\right ) \sin {\left (2 \pi t \right )}
\]