View Full Version : Laplace transform checking

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.

2015-Jul-11, 11:32 AM
Offhand, I'd say you found a typo, but I need a citation first! :)

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.

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:

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:

EETA: there does appear to be a fifth edition: