University:
Georgia Southern University
Course:
MATH 3230 | Ordinary Differential Equations
Academic year:

2024
Views:

280

Pages:

18
Author:

Jalen W.

Property 3: Exponential Factors Suppose that the Laplace Transform of f (t) exists for s > s0 , and call this F (s). Then, L{e at f (t)}(s) = F (s − a) for s > a + s0 . Proof This is because, by definition, Z ∞ Z at −st at L{e f (t)}(s) = e e f (t) dt = 0 ∞ e −(s−a)t f (t) dt. 0 If we let u = s − a, then we have Z ∞ L{e at f (t)}(s) = e −ut f (t) dt = F (u), 0 where u > s0 by the domain restrictions for F . Hence F (u) = F (s − a) and s − a > s0 implies s > s0 + a. Example Suppose f (t) = t 2 . Use the table of Laplace Transforms in the previous section to find F (s). Now, find the Laplace Transform of e 2t f (t). The Unit Step Function (AKA Heaviside function) The unit step function (or heaviside function) is a piecewise function defined by, ( 0, t < 0 u(t) = . 1, t ≥ 0 Then, using precalculus, ( 0, t < b u(t − b) = . 1, t ≥ b Example Find L{u(t − b)}(s). Property 4: Unit Step Function Factor Consider a function f (t) and suppose the Laplace Transform, F (s) exists for s > s0 . Then, L{u(t − b)f (t − b)}(s) = e −bs F (s), for s > s0 . Equivalently, we may say for a function g (t) whose Laplace Transform exists that L{u(t − b)g (t)}(s) = e −bs L{g (t + b)}(s). Proof Z ∞ L{u(t − b)f (t − b)}(s) = Z0 ∞ = e −ts u(t − b)f (t − b) dt e −ts f (t − b) dt b Let v = t − b, so that dv = dt. Z ∞ e −(v +b)s f (v ) dv 0 Z ∞ −bs = e e −vs f (v ) dv L{u(t − b)f (t − b)}(s) = 0 = e −bs L{f }(s). Example Find the Laplace transform of the following functions. 1. u(t − 1)e 2t−2 2. u(t − 1)e 2t 3. u(t − π2 ) sin(t − π2 ). 4. u(t − π2 ) sin(t). Example For the function f (t) below, find F (s)   1, 0 ≤ t < 1 f (t) = 0, 1 ≤ t < 2   2, t ≥ 2 A Box with Area 1 It may have occurred to you that we haven’t found any functions (yet) whose Laplace transform is equal to 1. Well, there is a function whose Laplace transform is equal to 1, but you aren’t going to like it. First let’s start off with the following definition. ( 0, t < 0 or t > ` . δ` (t) = 1 `, 0 ≤ t ≤ ` Then, the graph of δ` (t) is just y 1 ` x ` where the height of the box is area of the box is just 1. 1 ` and the width is just `. So the Example Compute the Laplace Transform of δ` (t − c). The Dirac Delta Function The Dirac delta function is defined to be ( 0, t = 6 0 δ(t) = lim δ` (t) = , `→0 ∞, t = 0 and the Laplace transform is defined to be L{δ(t)}(s) = lim L{δ` (t)}(s) = 1, `→0 for s > 0. In general, L{δ(t − c)}(s) = lim L{δ(t − c)}(s) = e −cs , `→0 for s > 0. Proof of Laplace Transform of the Dirac Delta Function L{δ(t − c)}(s) = lim L{δ` (t − c)}(s) `→0 1 − e −s` `→0 s` −s` se = lim e −cs `→0 s = lim e −cs = lim e −cs `→0 −cs = e Property 5: Dirac Delta Factor Consider the function f (t) and suppose the Laplace Transform exists for s > s0 . If f (t) is continuous at t = c, then L{δ(t − c)f (t)}(s) = e −cs f (c), for s > s0 . Proof The Mean Value Theorem Let f (x) be continuous on [a, b]. Then, Z a for some x ∗ ∈ [a, b]. b f (x) dx = f (x ∗ )(b − a) Proof L{δ(t − c)f (t)}(s) = lim L{δ` (t − c)f (t)}(s) Z ∞ = lim δ` (t − c)f (t) dt `→0 0 Z c+` 1 = lim f (t)e −st dt `→0 c ` 1 ∗ = lim f (t ∗ )e −st ((c + `) − c) `→0 ` `→0 for some t ∗ ∈ [c, c + `] by Mean Value Theorem. Proof L{δ(t − c)f (t)}(s) = = 1 ∗ lim f (t ∗ )e −st ((c + `) − c) ` ∗ lim f (t ∗ )e −st `→0 `→0 = f (c)e −sc since t ∗ ∈ [c, c + `] → {c} as ` → 0. Example Compute the Laplace Transform for the following: 1. δ(t) sin2 (24t) √ 2. δ(t − 1) t 2 + 4 3. δ(t + 1)e at 4. u(t − 1)δ(t − 1)

Section 8.2: Laplace Transforms of Special Functions

Section 8.2: Laplace Transforms of Special Functions - Page 3

Section 8.2: Laplace Transforms of Special Functions - Page 4

Section 8.2: Laplace Transforms of Special Functions - Page 5

Section 8.2: Laplace Transforms of Special Functions - Page 6

Section 8.2: Laplace Transforms of Special Functions - Page 7

Section 8.2: Laplace Transforms of Special Functions - Page 8

Section 8.2: Laplace Transforms of Special Functions - Page 9

Section 8.2: Laplace Transforms of Special Functions - Page 10

Section 8.2: Laplace Transforms of Special Functions - Page 11

Section 8.2: Laplace Transforms of Special Functions - Page 12

Section 8.2: Laplace Transforms of Special Functions - Page 13

Section 8.2: Laplace Transforms of Special Functions - Page 14

Section 8.2: Laplace Transforms of Special Functions - Page 15

Section 8.2: Laplace Transforms of Special Functions - Page 16

Section 8.2: Laplace Transforms of Special Functions - Page 17

Section 8.2: Laplace Transforms of Special Functions - Page 18

of 18

Details

Description

Struggling with your assignment and deadlines?

Free up your schedule!

Take 5 seconds to unlock