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DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t 2 0. Then the integralXflt)}e-stf(t) dtis said to be the Laplace transform of f, provided that ...

Question

DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t 2 0. Then the integralXflt)}e-stf(t) dtis said to be the Laplace transform of f, provided that the integral converges. Find {flt)}. (Write your answer as a function of $.) R9 = {: 0 <t < 1 t 2 1 {f(t)}(s > 0)

DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t 2 0. Then the integral Xflt)} e-stf(t) dt is said to be the Laplace transform of f, provided that the integral converges. Find {flt)}. (Write your answer as a function of $.) R9 = {: 0 <t < 1 t 2 1 {f(t)} (s > 0)



Answers

Determine the Laplace transform of the given function. $$f(t)=t^{2}, \quad 0 \leq t<2, \quad f(t+2)=f(t)$$

Let's find the laplace transform of our function F f t. So we C squared the laplace transform F of AIDS. It's equal to the integral from 0 to Infinity of t square E. Experience minus S t d C. And this is what our limits. As B approaches infinity. What do I have one over is cube minus S craig t squared minus two S t minus two E. Exponent minus S T. The interval is from server to be and this is going to give us two divided by sq yes, should be greater than so.

Let's find a laplace transform of our function f f t so we c squared. Allah please transform F of AIDS. It's equal to the integral from 0 to Infinity of t square E experience minus S t d C. And this is what our limits. As B approaches infinity. What do I have one over is cube minus S grade t squared minus two S t minus two E. Exponent minus S T. The intervals from server to be. And this is going to give us two provided by sq Yes, should be greater than so.

Hello. Divert todavia going to soar. Problem number I think Q f off the Israeli awareness Do u minus learned the whole square Indo your glove, yet it is shifted by pencil. It's only one thing. So the hardware, my ass learn here. So you bigger thing be in my Nesler my s one plus one 400 men sip into your birthday. So you're getting Dave my enough to plus my whole square into your lofty Wonderful A plus b the whole square flu B minus two the whole square Bless. Oh, do you mind us too? Plus one square into your true off day. So the U minus two the whole square into your lofty plus true dear minus two into your true off day one name Do your off day. So, Ray second shifting Theo Bigger, lackluster. I'm some off Dave Square. If it affected her by Yes, Kim. So it will be given us no effect earlier by ask you into intel apart by this to us less through window into the bar minus prove us. But yes, a square less. You didn't thought off minus two s never buy. Yes, there's that awful question. Thank you.

Okay, so I got the periodic function, which is defined as being eat the T between zero and one and its periodic with period. Big T is equal to one. So what is the LaPlace transform while from theorem 10.3 point three, the plaice transformers 1/1, minus E to the minus. Ask times period multiplied by the integral from zero to the period of each of the minus esti times by the function. So this becomes a very simple into girl. Andi, integrating this. We get minus this between zero and one. So evaluated at one. This is eat zero. So it's just one my minus one over us, minus one. And then that's plus because it's minus minus. Plus one over asked, minus one times. Sorry. Okay. So, yeah, we're evaluating T is equal to one and t Z zero. Right? So this is gonna be that and then minus, But then that's gonna be a plus. One over Asked, minus one. And that's it. Okay, so this is the answer to our problem. Okay. If we wish, we can combine it into a single common denominator. Okay? That's it.


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