Why Are My Laplace Transform Results Different?

[Drain Brain]

I came across this problem and solved it using different approach. I get a slightly different answers.

here's how it goes,

1st approach

$\mathscr{L}[te^{2t}\cos(3t)]$

first I get the laplace of something that's familiar to me which is $\mathscr{L}[t\cos(3t)]$

using this $\mathscr{L}[t^nf(t)]=(-1)^n\frac{d^n}{ds^n}\mathscr{L}[f(t)]$ $\mathscr{L}[t\cos(3t)]= -\frac{d}{ds}\left(\frac{s}{s^2+9}\right)$
=$ -\left(\frac{(s^2+9)(1)-s(2s)}{(s^2+9)^2}\right) = \frac{-s^2-9+2s^2}{(s^2+9)^2} = \frac{s^2-9}{(s^2+9)^2}$
shift:

$\mathscr{L}[te^{2t}\cos(3t)]= \frac{(s-2)^2-9}{\left((s-2)^2+9\right)^2}=\frac{s^2-4s+4-9}{\left(s^2-4s+4+9\right)^2} = \frac{s^2-4s-5}{\left(s^2-4s+13\right)^2}$

$\mathscr{L}[te^{2t}\cos(3t)]= \frac{s^2-4s-5}{\left(s^2-4s+13\right)^2}$***

2nd approach

First I get $\mathscr{L}[e^{2t}\cos(3t)]$

using $\mathscr{L}[e^{\alpha t}f(t)] = F(s-\alpha)$

$\mathscr{L}[e^{2t}\cos(3t)]$ --->>> $\mathscr{L}[\cos(3t)] = \frac{s}{s^2+9}|_{s-->s-2} = \frac{s-2}{s^2-4s+13} $

now using $\mathscr{L}[t^nf(t)]=(-1)^n\frac{d^n}{ds^n}\mathscr{L}[f(t)]$

=$ -\frac{d}{ds}\left(\frac{s-2}{s^2-4s+13}\right) = (s-2)\frac{d}{ds}(s^2-4s+13)^{-1}+(s^2-4s+13)^{-1}\frac{d}{ds}(s-2)$

= $(s-2)(-(s^2-4s+13)^{-2}(2s-4))+(s^2-4s+13)^{1}$

=$-(s^2-4s+13)^{-2}(2s-4)(s-2)+(s^2-4s+13)^{-1}$

=$-(s^2-4s+13)^{-2}(2s^2-8s+8)+(s^2-4s+13)^{-1}$

=$-\frac{2s^2-8s+8}{(s^2-4s+13)^2}+\frac{1}{(s^2-4s+13)}$

= $\frac{-2s^2+8s-8+s^2-4s+13}{(s^2-4s+13)^2}$

= $\frac{-s^2+4s+5}{(s^2-4s+13)^2}$
$\mathscr{L}[te^{2t}\cos(3t)]=\frac{-s^2+4s+5}{(s^2-4s+13)^2}$ ***

I can't figure out why they were different please help me to spot the error.

thanks!I think I found it! arghhh I'm kind of sleepy I missed the -1 on the 2nd method!(Headbang)(Headbang)

 

[jvicens]

Hi there,

Thank you for sharing your approaches to solving this problem. It seems like you have already identified the error in your second approach - missing the negative sign. This can often happen when we are working on problems late at night or when we are tired. It's important to always double check our work and make sure we haven't made any simple mistakes.

In terms of why the two approaches give slightly different answers, it could be due to rounding errors or slight differences in the methods used. However, both of your final answers seem to simplify to the same expression, so they are essentially equivalent.

In any case, it's great that you were able to spot the error and correct it. This is an important skill for scientists - being able to recognize and correct mistakes in our work. Keep up the good work!