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5. Solve the following DEs by the Laplace transform method:. (a) y" + 5y' + 4y = 0 y(0) =1, Y(0) = 0 (b) y" - 6y' + 9y = e A* y(O) =0, Y() =0...

Question

5. Solve the following DEs by the Laplace transform method:. (a) y" + 5y' + 4y = 0 y(0) =1, Y(0) = 0 (b) y" - 6y' + 9y = e A* y(O) =0, Y() =0

5. Solve the following DEs by the Laplace transform method:. (a) y" + 5y' + 4y = 0 y(0) =1, Y(0) = 0 (b) y" - 6y' + 9y = e A* y(O) =0, Y() =0



Answers

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$y^{\prime \prime}-4 y^{\prime}+5 y=0, y(0)=1, y^{\prime}(0)=2$$

So for Why? Why you keep it, you five. And why weren't three year old hold Teoh One? So that'll be rearranged This plus tree Why, right? I notice one when you came Why? Who follows before? Why has equal to by s Square one Waas wanted to buy by this queer was four. Why I fine to t plus.

All right. So our problem here is wise old prime plus two. I promptly fired White zero. Were given that Y zero. It was too with my prime zero, negative one started the applause transform the S squared why uh minus? It's going to be s times Y zero which means to me to us and it's going to be minus why privacy is negative one plus one. Yeah, plus two. As why are we going to be minus two times Y zero in minus for just five classrooms from why eagles here. Alright, separate them out. S square plus two S plus five, remove all the other terms to the right hand side. This can be to S minus one plus four, so plus three. Not to get the questions were alone will do pollsters from Y to go to to S Plus three all over S squared plus two S plus five. Okay. And now from here it kind of looks like we can work this into a form that we recognize and try to make this into the past transform of the exponential function as a senior sort of function. So first we're on now here around two F plus three all over. And then if you look here, you see the S square plus to us, you know, that's the beginning to the function as plus one squared, which we know will come out to s squared plus two S plus one. But we need to plus five. We're going to add another four. Now if you carry on. But the bombs we're gonna have to outside of S plus one. You need that up top for this for this form we're going for. Uh And then we need to get a plus three but it's going to get us to s plus to add another one. Here There you have s. Plus one squared plus two square. So I'm gonna write this two squared because in a loss transform for next bunch of function times as high as soil function. Um This term here is has to be a square. So now we're gonna separate this out. We'll get that too. Times S-plus one all over. S plus one squared plus two squared plus one over S plus one squared plus to start. All right? But here on the on this side here we know that these terms need to match. So rewrite this to S plus one over S plus one squared plus two squared Plus we'll take out 1/2 to make that equal. Well we have to up here S plus one squared plus two squared. So now we can solve especially why is he going to the applause transform of took two out. So I forgot uh S. Plus one over S. Plus one squared. Kind of messed it up. Yeah. Yes plus one squared plus two squared. Uh Plus one half. Universal boss transformed of two over S plus one squared plus two straight. Uh What we know from Y. Equals to eat a negative T. Uh co sign of two T. Plus one half E. To the negative T. Sign of two T. That is the answer to that bro.

In the problem we have four wide or less Plus four. White as my plus five. Why that is zero Or Y. of zero is equal to. Wanna. Why does zero is equal to -1 upon to? So we have four into it is quite lap lots of hawaii minus is Y zero minus minus zero plus four into S. Lab plus Y minus Y zero plus five plus. Why? That equals zero. Our father we have for esquire lap plus white minus photos plus two plus forest, lap plus white minus four plus five lap plus Y. That is each call to zero. Hence we have lab plus Y. Into for esquire plus forest plus five. That equals for S. Plus two. So laplace of Y. Is equal to forest plus two upon To s plus one Whole Square Plus two. Police question. Now further we have Y. F. D. Equal laplace in brasov. Forest Plus two upon To S Plus one. All is quiet plus two square. So further we have Y. F. D. Equal to to into into s. Plus one upon to bon forward into S. Plus gonna 0.2 whole square plus four. This is equal to S. Plus half. He wanted to buy. Yes plus on upon two whole square plus one. So this is our lab plus in verse and this is our last lesson worse. Therefore, why have they become? You're the power minus T. Upon to costume. So this is the answer.

Himself with speculation. Like why don't time my? Why equals zero And you're right. This is general question. Why? CEO on why prime off zero using all possibly in Italy s school aired remain. It's one multiplied by lie. Wait, I zero five time syrup So why has supposed to Esther names? Why here? Uh equal to which is he quarter? Why here? Why Prime put zero by two time. Do you know is put one plus my prime here? Why? Estimates says difficult. Why using injuries it b y t c one.


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