### Question 1

*Find the Laplace transform of the following functions.*

### (i)

#### Solution

Using linearity:

.

Now using the tables:

.

### (iii)

.

#### Solution

This needs the First Shift Theorem, which states:

,

where . Now looking at what we have to transform:

,

clearly what we need to do is find the Laplace transform of , — and then replace by . Now the Laplace transform of , by the tables, is . Hence

.

## Question 2

*Find the Laplace transforms of the functions that satisfy the following differential equations.*

### (i)

, .

#### Solution

Take the Laplace transform of both sides:

.

Now use linearity;

.

Now consulting the tables (for the first differentiation theorem and the laplace transform of 1):

.

Now use the boundary condition — ;

Solving for ;

.

### (ii)

; , .

#### Solution

Taking the Laplace transform of both sides (using linearity and the fact that the laplace transform of is — also note that ):

.

Now applying the differentiation theorems:

.

Applying the boundary conditions:

.

Now all that remains is to solve for :

.

## 2 comments

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May 19, 2011 at 7:47 pm

Graham KielyHi JP,

Any chance you could post the solutions to the Laplace transforms of “2 sin 3t cos t” , “3 cos 4t sin 2t” just so I/we have a couple of more examples of using the (A+B) formula in a product of sin and cos situation.

Thanks,

Graham

May 20, 2011 at 4:33 pm

J.P. McCarthyRoger.

O.K., the main principle is that the Laplace transform is linear — that is it handles sums — we can split sums and take out constants. Therefore if we can write terms of the form , etc. as sums of sines and cosines we are away with it.

The formulae that convert products like these into sums is in the log tables. Taking the example of, say, , we use the formula

We can write . Hence

.

Now we can use the linear nature of the Laplace transform:

Now using the tables to find the Laplace transforms of terms of the form and :