Section 4.1 Laplace Transforms & Inverse Transforms

Definition: Given a function f (t) defined for all t ≥ 0, the Laplace Transform of f is This will produce a function of s, and we shall denote this function F(s). (i.e. L{ f (t) } = F(s))

Ex. 1 Compute the Laplace transform of f (t) = e t

Ex. 2 Compute the Laplace transform of f (t) = e At

Which functions have Laplace transforms and which don't?

f (t) F(s) 1 t t 2 t 3 t 4 t 5 t 1/2

We can express these Laplace transforms using a non-elementary function called the gamma function:

Ex. 3 Compute L{ t α } and express the answer in terms of the gamma function.

Some information regarding Γ: Γ(1) = 1 Γ(x + 1) = x Γ(x) Γ( 1 / 2 ) =

Some information regarding Γ: Γ(1) = 1 Γ(x + 1) = x Γ(x) Γ( 1 / 2 ) = So, if α is an integer then

The unit step function:

More generally:

Ex. 4 Compute L{ u(t) }

Ex. 5 Let A > 0. Compute L{ u(t – A) }

Theorem: L{ c f (t) } = c L{ f (t) } for any constant c. L{ f (t) + g(t) } = L{ f (t) } + L{ g(t) } L{ f (t) – g(t) } = L{ f (t) } – L{ g(t) }

Ex. 6 Let k > 0. Compute L{ cosh(kt) }

Ex. 7 Let k > 0. Compute L{ cos(kt) }

We also have: (for s > k) and (for s > 0)

Inverse Laplace Transforms: L{ f (t) } = F(s) ⇕ L –1 { F(s) } = f (t) Examples:

Ex. 8 Compute the following inverse Laplace transforms: (a) (if n is an integer)

Ex. 8 Compute the following inverse Laplace transforms: (b)

Ex. 8 Compute the following inverse Laplace transforms: (c)

Ex. 8 Compute the following inverse Laplace transforms: (d)

Ex. 8 Compute the following inverse Laplace transforms: (d)

Section 4.2 Transformation of Initial Value Problems

Theorem: L{ f ′ (t) } = s L{ f (t) } – f (0) = s F(s) – f (0)

Ex. 1 Suppose L{ f (t) } = F(s). Use this last theorem to find L{ f ″(t) } and L{ f (3) (t) }.

Theorem: L{ f (n) (t) } = ___________________________________________________

Ex. 2 Use Laplace transforms to solve the following initial value problem: y″ + 2y′ – 8y = 0, y(0) = 4, y′(0) = –10

Ex. 3 Use Laplace transforms to solve the following initial value problem: x″ + 3x′ + 2x = t, x(0) = 0, x′(0) = 2

Ex. 3 Use Laplace transforms to solve the following initial value problem: x″ + 3x′ + 2x = t, x(0) = 0, x′(0) = 2

Section 4.3 Translation and Partial Fractions

Problem: Try to use Laplace transforms to solve the following: y″ + 6y′ + 34y = 0, y(0) = 3, y′(0) = 1. Answer: L{ y″ + 6y′ + 34y } = L{ 0 } (s 2 F(s) – 3s – 1) + 6(sF(s) – 3) + 34F(s) = 0 (s 2 + 6s + 34) F(s) = 3s + 19

Theorem: (Translation on the s-Axis) If A is a constant and L{ f (t) } = F(s) then L –1 { F(s – A) } = e At f (t)

Theorem: (Translation on the s-Axis) If A is a constant and L{ f (t) } = F(s) then L –1 { F(s – A) } = e At f (t) In particular:

Theorem: (Translation on the s-Axis) If A is a constant and L{ f (t) } = F(s) then L –1 { F(s – A) } = e At f (t) Note that this theorem also implies that L{ e At f(t) } = F(s – A)

Ex. 1 Find the inverse Laplace transform of

Ex. 2 Solve y″ + 6y′ + 34y = 0, y(0) = 3, y′(0) = 1

Section 4.5 Periodic & Piecewise Continuous Input Functions

Recall:

Ex. 1 A graph of f (t) is given below. Give a formula for the function f (t) in terms of step functions.

Ex. 2 A graph of g(t) is given below. Give a formula for the function g(t) in terms of step functions.

Ex. 3. h(t) can be rewritten as u(t – A) f (t – A) for some constant A and some function f (u). Find A and f (u).

Theorem: If L{ f (t) } = F(s) then L{ u(t – A) f (t – A) } = e – As F(s)

Ex. 4. Compute L{ g(t) }

Theorem: If f (t) is a periodic function with period p then

Ex. 5 The graph of f (t) is given below. Determine the Laplace transform of f (t).