Engineering Mathematics Class #11 Laplace Transforms (Part1) Sheng-Fang Huang

Solving an IVP by Laplace transforms The Laplace transform method is a powerful method for solving linear ODEs and corresponding initial value problems, as well as systems of ODEs arising in engineering.

6.1 Laplace Transform. Inverse Transform. Linearity. s-Shifting ƒ(t) is a function defined for all t ≥ 0. Its Laplace transform, , is denoted by F(s), which is (1) Here we must assume that ƒ(t) is such that the integral exists (that is, has some finite value).

Inverse Transform The given function ƒ(t) in (1) is called the inverse transform of F(s) and is denoted by ; that is, (1*) Note that (1) and (1*) together imply = ƒ, and = F.

Example 1: Laplace Transform Let ƒ(t) = 1 when t ≥ 0. Find F(s). Solution. From (1) we obtain by integration The interval of integration in (1) is infinite. Such an integral is evaluated according to the rule

Example 2 Laplace Transform of the eat Let ƒ(t) = eat when t ≥ 0, where a is a constant. Find Solution.

Linearity of the Laplace Transform THEOREM 1 The Laplace transform is a linear operation; that is, for any functions ƒ(t) and g(t) whose transforms exist and any constants a and b the transform of aƒ(t) + bg(t) exists, and

Example 3 Application of Theorem 1: Hyperbolic Functions Find the transforms of cosh at and sinh at. Solution. Since coshat = 1/2(eat + e-at) and sinhat = 1/2(eat – e-at),

Example 4 Cosine and Sine Derive the formulas Solution:

By substituting Ls into the formula for Lc on the right and then by substituting Lc into the formula for Ls on the right, we obtain Solution by Transforms Using Derivatives. See next section.

Solution by Complex Methods Solution by Complex Methods. In Example 2, if we set a = iω with i = (–1)1/2, we obtain Now by Theorem 1 and eiωt = cos ωt + i sin ωt If we equate the real and imaginary parts of this and the previous equation, the result follows.

Some Functions ƒ(t) and Their Laplace Transforms

s-Shifting: Replacing s by s – a in the Transform First Shifting Theorem, s-Shifting THEOREM 2 If ƒ(t) has the transform F(s) (where s > k for some k), then eatƒ(t) has the transform F(s – a) (where s – a > k). In formulas, or, if we take the inverse on both sides,

Example 5 s-Shifting: Damped Vibrations. Completing the Square From Example 4, For instance, use these formulas to find the inverse of the transform