View Full Version : Laplace transform checking

Glom

2015-Jul-11, 08:06 AM

I find myself disagreeing with a book on this particular case.

It begins with the harmonic oscillator, whose position at time t is given by:

af''(t) + bf(t) = 0, f(0) = \alpha, f'(0) = \beta

Taking Laplace transform:

L{af''(t) + bf(t)} = L{0}

a[s2F(s) - s\alpha - \beta] + b[F(s)] = 0

(as2 + b)F(s) = a(s\alpha + \beta)

But the book says, (as2 + b)F(s) = s\alpha + \beta. No a on the RHS.

grapes

2015-Jul-11, 11:32 AM

Offhand, I'd say you found a typo, but I need a citation first! :)

Glom

2015-Jul-11, 11:50 AM

Advanced Engineering Mathematics Fourth Edition by K. A. Stroud

Page 131

Ken G

2015-Jul-11, 01:28 PM

Your math shows that it is a typo, but if one is not convinced a good thing to do is to check the answer physically. One does that by testing the limits. A good way to do that is to set b=0, for then we simply have no spring, and we must have motion at constant speed: f(t) = alpha + t*beta, independent of a (where of course a is the mass). So Newton tells us that if there is no spring, all masses with the same initial data move the same. Without even calculating any Laplace transforms, I can see that your solution has that property, and theirs doesn't.

grapes

2015-Jul-12, 11:52 AM

Advanced Engineering Mathematics Fourth Edition by K. A. Stroud

Page 131

I found an online copy of that edition:

https://nieeeslibrary.files.wordpress.com/2013/01/advanced-engineering-mathematics-4th-ed-k-stroud.pdf

It's not just a typo, it's worse! :)

For instance, the example 2 following (page 132, p.157 of the PDF) actually uses that erroneous formula to say that f'(0) = 4, in the conditions, and they arrive at the following function which clearly has f'(0)=4/5:

f(t) = \frac{2\sqrt{2}}{5} sin{\sqrt{2}t}

Surely this must be in some errata somewhere!

ETA: somebody else found the same error, they had no luck in the errata:

http://math.stackexchange.com/questions/721241/are-these-laplace-transforms-wrong-in-strouds-advanced-engineering-math-book

EETA: there does appear to be a fifth edition:

http://www.amazon.com/gp/aw/d/0831134496/ref=pd_aw_fbt_14_img_2?refRID=005EQ19ZV85AP6RSFP0E

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