Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function:x' + x = 4+ 8(t – 1),x(0) = 0.In the following parts, use h(t – c) for the Heaviside function he(t) if necessary.a. Find the Laplace transform of the solution.L{r(t)}(s) =b. Obtain the solution x(t).x(t) =%3Dc. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 1.if 0

Given differential equation is,x1+x=4+δ(t-1) , x(0)=0 -(1)We know that,L{x'(t)} =L{x(t)}-…

Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function: x' +x = 4+ 8(t – 1), x(0) = 0. In the following parts, use h(t - c) for the Heaviside function h.(t) if necessary. a. Find the Laplace transform of the solution. L{x(t)}(s) = b. Obtain the solution r(t). x(t) = c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 1. if 0

Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function: x' +x = 4+ 8(t – 1), x(0) = 0. In the following parts, use h(t - c) for the Heaviside function h.(t) if necessary. a. Find the Laplace transform of the solution. L{x(t)}(s) = b. Obtain the solution r(t). x(t) = c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 1. if 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function:
x' + x = 4+ 8(t – 1),
x(0) = 0.
In the following parts, use h(t – c) for the Heaviside function he(t) if necessary.
a. Find the Laplace transform of the solution.
L{r(t)}(s) =
b. Obtain the solution x(t).
x(t) =
%3D
c. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 1.
if 0 <t < 1,
a(t) =
%3D
if 1 <t < o.

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