Question

need help with laplace transform stuff...

Answer 1

here is the function that I need to find the laplace for:
\[f(t)=4t ^{2}-5\sin3t\]

Answer 2

i did the integration all the way up until the point where i get the same integral of the 2nd term in the given function above.

Answer 3

how should i go about solving for the final answer?

Answer 4

here is my incomplete so far:
\[\left[ \frac{ -4t ^{2}e ^{-st} }{ s } -\frac{ 2te ^{-st} }{ s ^{2} }-\frac{ 2e ^{-st} }{ s ^{2} }+\frac{ 5\sin3te ^{-st} }{ s }+\frac{ 15\cos3te ^{-st} }{ s ^{2} }\right]_{0}^{\infty}+(\frac{ 45 }{ s ^{2}) \int\limits_{0}^{\infty}}\]

Answer 5

answer^

Answer 6

@dan815

Answer 7

the 2nd part should read:
\[\int\limits_{0}^{\infty}\frac{ 45\sin3te ^{-st}dt }{ s ^{2} }\]

Answer 8

@Preetha

Answer 9

@thomaster

Answer 10

@jim_thompson5910 @Hero

Answer 11

That's Wayyy too long...haha you did it wrong

Answer 12

haha well thanks for pointing that out! what should I do then?

Answer 13

Do you know the Laplace Transform for 4t^2?

Answer 14

never mind I think I got it.

Answer 15

It's just two very simple laplace transforms.

Answer 16

have to do it the long way, not by using the table.

Answer 17

but thanks though

Answer 18

lol..

Answer 19

no wait...

Answer 20

how am I wrong?

Answer 21

@FutureMathProfessor

Answer 22

im not finished yet, so probably that's why it looks wrong.

Answer 23

I just need help with the last loopy part

Answer 24

so do i just subtract it from both sides?

Answer 25

@Mathguy1234 any idea?

Answer 26

Yeah. The laplace transform of 4sin3t is just 4a/(s^2+a^2)

Answer 27

i understand what u r trying to say, but can u help with the integral bit?

Answer 28

I can.one. Sec

Answer 29

do u think i should break them up and solve them separately?

Answer 30

or does that even matter?

Answer 31

I think the a=3 in that case?

Answer 32

Laplace Transforms were the only thing I was good at in differential equations class :evillaughknowingigottatakeitagain:

Answer 33

can we forget about the table for a little bit, please?

Answer 34

haha @FutureMathProfessor

Answer 35

sorry to hear that @FutureMathProfessor

Answer 36

LOL its ok, I think I'll do better after I test out of linear algebra! xD

Answer 37

Okay this is how to do it.
1. The laplace transform of af(x) + bg(x) is a times the laplace transform of f(x) + b times the laplace transform of g(x).
2. The lAplace transform of t^2 is 2/s^3 and that of sin3t is 3/(s2+9)
3. If you want the proof from first principles here's a good site http://www.khanacademy.org/math/differential-equations/laplace-transform/laplace-transform-tutorial/v/laplace-transform-3--l-sin-at

Answer 38

khanacademy is too slow for my taste, but thanks though!

Answer 39

im trying to proof it myself, but just having troubles with the calc part.

Answer 40

Cause typing it out is a bit hairy especially cause I'm on my ipad

Answer 41

no prob, thanks anyways!

Answer 42

Fine

Answer 43

im just going to break the two and solve from there.

Answer 44

How do you show math here

Answer 45

You just need integration by parts for the second one.

Answer 46

yes, I know that...I did it twice, but wasn't sure as to how to write the final answer. I will try the way I am doing it and then, see if I get the same answer as I would from the table.

Answer 47

@UnkleRhaukus

Answer 48

@UnkleRhaukus how should I finish off the integration to get the final answer where I can take the limits? any idea?

Answer 49

@FutureMathProfessor I think it's a good idea that u r taking linear algebra. very useful in diffy eqs.

Answer 50

use a table for this problem , there is no need to integrate

Answer 51

no he have to use the theorem

Answer 52

we*

Answer 53

\[f(t)=4t^2-5\sin(3t)\\
\,\\
\mathcal L\{f(t)\}=4\mathcal L\{t^2\}-5\mathcal L\{\sin(3t)\}\]

Answer 54

ok?

Answer 55

I did do that...I am at the part where I am getting the same integral as my 2nd integral term.

Answer 56

i want to refresh my memory as to how to deal with that...i know u can add that term to both sides and stuff, but am not sure if i can here.

Answer 57

\[\large \boxed{\mathcal L\{t^n\}=\dfrac{n!}{s^{n+1}}}\\
\large \boxed{\mathcal L\{\sin(nt)\}=\dfrac{n}{s^2+n^2}}\]

Answer 58

\[\begin{align*}
\mathcal L\left\{ t^{n}\right\}&=\int\limits_0^\infty t^{n}e^{-{s}t}\text dt
\\\text{let } t=\frac u{s}\qquad\quad\text dt=\frac{\text du}{s}&
\\t=0\rightarrow u=0\qquad t=\infty\rightarrow u=\infty&
\\&=\int\limits_0^\infty \left(\frac u{s}\right)^{n}e^{-{s}\frac u{s}}\frac{\text du}{s}
\\&=\frac{1}{{s}^{{n}+1}}\int\limits_0^\infty u^{n}e^{-u} {\text du}\\
\\&=\frac{\Gamma({n}+1)}{{s}^{{n}+1}}
\\&=\frac{{n}!}{{s}^{{n}+1}},& {n}\in\mathbb Z\geq0
\end{align*} %unk
\]

Answer 59

one way to prove the laplace transform of a sine function is to use complexification\[\begin{align}
\mathcal L\left\{ \sin({n}t)\right\}&=\int\limits_0^\infty \sin({n}t)e^{-{s}t}\text dt\\
\\&=\int\limits_0^\infty {\frak I}\left(e^{i{n}t}\right)e^{-{s}t}\text dt\\
\\&={\frak I}\left(\int\limits_0^\infty e^{i{n}t}e^{-{s}t}\text dt\right)\\
\\&={\frak I}\left(\int\limits_0^\infty e^{-({s}-i{n})t}\text dt\right)\\
\\&= {\frak I}\left(\left.\frac{e^{-({s}-i{n})t}}{-({s}-i{n})}\right|_0^\infty\right)\\
\\&= {\frak I}\left(\frac{1}{{s}-i{n}}\right)\\
\\&= {\frak I}\left(\frac{1}{{s}-i{n}}\times\frac{{s}+i{n}}{{s}+i{n}}\right)\\
\\&= {\frak I}\left(\frac{{s}+i{n}}{{s}^2+{n}^2}\right)\\
\\&= \frac{n}{{s}^2+{n}^2}\\
\end{align}%unk\]

Answer 60

does that help?

Answer 61

..?

Answer 62

not the 2nd part.

Answer 63

Are you still having trouble with your question?

Answer 64

can u do it any other way besides complexification b/c I never heard of it until now.

Answer 65

@UnkleRhaukus

Answer 66

You want me to prove it right?

Answer 67

yes please if u can.

Answer 68

Yeah, I can.

Answer 69

im stuck at the end part where I get the same laplace of sin3t part.

Answer 70

Pay attention; I'll be using basic integration.
But why don't you just check tables?

Answer 71

i did.

Answer 72

i'm suppose to get -15/s^2+9

Answer 73

yeah, that's the answer.

Answer 74

also, i have to prove it.

Answer 75

Okay, to prove the genarl case of asin(bt) or just 5sin3t?

Answer 76

*general

Answer 77

the later would be better.

Answer 78

so you have to prove \[ \boxed{\mathcal L\{\sin(nt)\}=\dfrac{n}{s^2+n^2}}\]
and you dont like the complexification method , ?

Answer 79

well i never heard of it so, no use.