The DE Tutorial is currently being split up into different threads to make editing these posts easier.

Laplace Transforms (Part I - Introduction, IVPs and Partial Fraction Techniques)

There are many types of transformations out there. For example,differentiationandintegrationare types of linear transformations. However, there is one particular transform that we would like to analyze. This transform is of the form:

where is called thekernelof the transformation.

In this case, we are interested in the transform with a kernel of . With this kernel, we take and transform it into another function . This transformation described by is called theLaplace Transform. It is denoted by .

Before we go and derive all the common Laplace Transforms (we will derive many more as we get futher into later posts), let us take a look at a familar function to some of us (this may also be totally knew to some of you out there).

Given , where , we define theGamma Function. It has the property and .

Now, if , then it follows by a similar idea that . If we continue simplifying, we have

This implies that when , .

(Thus it is interesting to point out that since , an identity for factorials.)

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Common Laplace Transforms

In this part, I will list the common Laplace Transforms, and leave the derivation of each in a spoiler for you to look at if you decide too.

Spoiler:

(This will pop up again, when we talk about translation theorems)

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; If

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Given

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Given

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Given

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Given

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Given

Spoiler:

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Let us go through some examples on how to apply linearity and some of these formulas.

Example 31

Find the Laplace Transform of

By linearity, we have

Taking into consideration Gamma function properties, we have .

Its not hard to show that . Therefore, .

Thus,

Example 32

Find the Laplace Transform of

Note that

Therefore,

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Inverse Laplace Transforms

As the name suggests, the Inverse Laplace Transform applied to a function will give you the original :

We now list the common inverse Laplace Transforms:

It is also worth mentioning that the Inverse Laplace Transform is linear.

Let us now go through a couple examples.

Example 33

Find the Inverse Laplace Transform of

Example 34

Find the Inverse Laplace Transform of

Example 35

Find the Inverse Laplace Transform of

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Laplace Transforms and IVPs (involving Partial Fraction Techniques)

We now introduce a method of solving initial value problems with Laplace Transforms. Before we go through this method, we first need to find the Laplace Transforms for , , and in general

I will leave the derivation of each in a spoiler.

Spoiler:

Spoiler:

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Translation Theorem

We will discuss an important translation theorem:

Theorem: If exists for , then exists for and .

Pf: Its obvious that

As a result of this translation theorem, we have six more Laplace Transforms to add to the list (I leave it for you to verify them):

There is one more interesting Laplace Transform worth considering:

With these fundamental Laplace Transforms, we can now tackle some initial value problems (some of these may require partial fraction techniques).

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Example 36

Use Laplace Transforms to solve the IVP

First, we take the Laplace Transform of both sides:

Applying the proper formulas and translations, we have

Now apply the initial conditions and to get

Now here comes the fun part: Take the Inverse Laplace transform of both sides to find the solution .

Lets consider each fraction individually.

First, consider .

To help us find the Inverse Laplace Transform, we need to apply partial fractions (I will redo this problem in the next post, when I talk aboutconvolution):

.

Our objective now is to find A, B, C, and D.

First, multiply both sides by the common denominator to get

If we take , we have

.

If we take , we have

If we take , we have

If we take , we have

This simplifies to

Thus, and

Thus,

Therefore,

Note that

Therefore, we finally have

Now, we need the second half of the solution! (We have only part of it!) XD

We now consider the other Inverse Laplace Transform:

We see that

Therefore, we now see that

Example 37

Use Laplace Transforms to solve the IVP

First apply the Laplace Transform on both sides to get

Applying the initial conditions and , we have

This is where the Laplace Transform of an Integral comes into play nicely (to avoid partial fractions)

In finding , we see that

Therefore,

Now, .

Therefore,

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This will conclude the first post on Laplace Transforms. I'm not sure when I will be able to post again, now that I start classes today. I'll try to find some time in the next several weeks to do so.