The issue here is that we have no product rule or chain rule for doing Laplace Transforms. The Laplace Transform is, however, linear and this means that we are good at handling sums.

Therefore if we could write as a sum then we would be away with it. Now there are two ways to do that. One is to use the Binomial Theorem but at this stage if you don’t know what that is you should just multiply out as follows:

.

Now we can transform all of these using

and

which is on the Laplace tables. Thus we have

.

Regards,

J.P.

If u get a chance could u show me how to get the laplace transform of .

Thanks

]]>,

morryah,

.

Regarding question 3, the answer given there is certainly wrong… see here https://jpmccarthymaths.com/2013/03/23/math6037-weeks-7-8/#comment-507

Regards.

J.P.

Many thanks.

]]>No bother. You can use Maple to check answers also.

Q. 1(a) — correct

Q. 1(b) — correct

Q. 1(c) (i) — is it not ?

Q. 1(c) (ii) — correct

Q. 1(c) (iii) — not quite. A partial fraction expansion gives

. You transformed the second term incorrectly.

Q. 1(d) — correct.

Q. 1(d) — I don’t know what happened to you here… The answer is 0.8862.

Q. 2(a) — Your anti-derivative is correct but not the evaluation. The answer is .

Q. 2(b) — hard yes but important if you are progressing on

Q. 2 (c) (i) — first shift theorem . However the first shift theorem gives

.

Q. 2 (c) (ii) — use the identity .

Q. 3 (a) — should it be of that? The lack of a cosine term is fine and will always happen when (why?)

Q. 3 (b) — mighty stuff considering that we never covered it in class!! You have it sussed out anyway… take your equation and ‘divide’ by and that is basically it.

Q. 4 (a) (i) — write it as and do two Chain Rules.

Q. 4 (a) (ii) — correct.

Q. 4 (b) — correct

Q. 4 (c) — correct

Regards,

J.P.

No detail needed – yes or no is fine – Ill go back to the drawing board if wrong.

Q1(a).

Using

.

comparing coefficients I get

Q1(b)

Completing the square I get . I work on getting a shifted cosine & as a result a shifted sine by +/- 3 then * & / 4

Leaving me with of the above giving me

Q1(c)(i)

by working out and dividing each term by

Although probably longer can this be done by saying the denominator is and putting a constant over each and compare coefficients. It works but would it in all cases. Of course it only works if the answer is right.

Q1(c)(ii)

by splitting and working on separate parts

Q1(c)(iii)

Q1(d)

from

Q1(e)

351.9346 using 0.25,0.75,1.25,1.75,2.25,2.75

Q2 (a)

13.40 by evaluating from from 0 to 2.

Q2 (b)

Using the definition as instructed ended up with

Took me a while to work through it with a lot of workings – made me contemplate not doing question 2. You make it look too easy in notes for winter 2012 so maybe just more practice needed here

Q2 (c)

Struggled here – brain was tired after 2b – any tips??

Q3 (a)

worked from completing the square of

From advice worked with calculating and where stating underdamping.

Looking at your expected answers was slightly concerned not to see a .

How do you determine the period & duration of oscillation? – Just for info

Q3 (b)

510 mm/s from

Q4 (a) (i)

Not attempted this exercise

Q4 (a) (ii)

Q4 (b)

5.77 @ 0.9 working with

Q4 (c)

1.3072

Appreciate any feedback.

]]>