Simplifying a Laplace Transform

[Steve Macwell]

Hello, I'm a little confused with how the final algebraic manipulation works out with this Laplace transformation which is:

p(t) = Pt*e^(-t/T) (P and T are constants)

Using the result of te^(-at) = 1/(s+a)^2

So I let a = 1/T, which gave me:


P/(s+1/T)^2

However, the mark scheme showed the result as:

1.) T^2/(sT+1)^2, which finally gives:


2.) PT^2/(sT+1)


I don't understand what has actually happened on the last two steps? Just in case I have not been specific I've labelled the two steps 1&2. These are the steps I need an algebra master to take a look at, and help me to understand if possible.

Thanks.

 

[DrPhil]

Hello, I'm a little confused with how the final algebraic manipulation works out with this Laplace transformation which is:

p(t) = Pt*e^(-t/T) (P and T are constants)

Using the result of te^(-at) = 1/(s+a)^2

So I let a = 1/T, which gave me:


P/(s+1/T)^2

However, the mark scheme showed the result as:

1.) T^2/(sT+1)^2, which finally gives:


2.) PT^2/(sT+1)


I don't understand what has actually happened on the last two steps? Just in case I have not been specific I've labelled the two steps 1&2. These are the steps I need an algebra master to take a look at, and help me to understand if possible.

Thanks.

The algebra to get to (1) is just to clear fractions by multiplying both numerator and denominator by T^2.

\(\displaystyle \displaystyle \dfrac{1}{\left(s + \frac{1}{T}\right)^2} \times \dfrac{T^2}{T^2} = \dfrac{T^2}{(sT + 1)^2}\)