Question

give me trick how to solve any Laplace transform equation

Answer 1

The behavior of a laplace-transformed function F(s) as s->infinity depends on the function's behavior as x-> 0. For example, functions that don't decay near x=0, such as f(x)=1, f(x)=cos(x), f(x)=cosh(x), f(x)=e-x , decay like 1/s as s gets large. Functions that decay lineary near x=0, such as f(x)=x, f(x)=sin(x), or f(x)=sinh(x) , decay like 1/s2 as s gets large. The general relation being: if f(x)->0 faster as x->0, then F(s) has faster decay near s=infinity. You might also want to think about what happens when f(x) is identically zero near x=0.

Answer 2

As we saw in the last section computingLaplace transforms directly can be fairly complicated.  Usually we just use a table of transforms when actually computingLaplace transforms.  The table that is provided here is not an inclusive table, but does include most of the commonly used Laplace transforms and most of the commonly needed formulas pertaining toLaplace transforms.

 

Before doing a couple of examples to illustrate the use of the table let’s get a quick fact out of the way.

 

Fact

Given f(t) and g(t) then,

                                                  

for any constants a and b.

 

In other words, we don’t worry about constants and we don’t worry about sums or differences of functions in takingLaplace transforms.  All that we need to do is take the transform of the individual functions, then put any constants back in and add or subtract the results back up.

 

So, let’s do a couple of quick examples.

 

Example 1  Find the Laplace transforms of the given functions.

(a)    [Solution]

(b)    [Solution]

(c)    [Solution]

(d)    [Solution]

 

Solution

Okay, there’s not really a whole lot to do here other than go to the table, transform the individual functions up, put any constants back in and then add or subtract the results.

 

We’ll do these examples in a little more detail than is typically used since this is the first time we’re using the tables.

 

(a)