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V" + y (0) +0 ( - 2) Q3 (15 pts: ) Solve the WVP % (0) = 0,y' (0) = 0 Do not use the convolution integral}(4) _Q4 (20_5+5+5+5 pts: _...

Question

V" + y (0) +0 ( - 2) Q3 (15 pts: ) Solve the WVP % (0) = 0,y' (0) = 0 Do not use the convolution integral}(4) _Q4 (20_5+5+5+5 pts: _

V" + y (0) +0 ( - 2) Q3 (15 pts: ) Solve the WVP % (0) = 0,y' (0) = 0 Do not use the convolution integral} (4) _ Q4 (20_5+5+5+5 pts: _



Answers

Solve the given initial-value problem up to the evaluation of a convolution integral. $$y^{\prime \prime}+y=e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=1$$

In the problem by this question. So we have a date. The fabulous. So taking the Nablus off Why don't last minus two by naso Lasted one equal to is far. Why, yes. My name is why you know minus white asked. You know equal This is my ass to India is why is minus Why do you equal is over a ceasefire Class four You had prosperity so loveless is is for now lease is this fire? Why is this is zero on this is one So it is minus one minus two into is why is my ass you know And here we have the 1910 Why it is so blessed in why yes equals do is over inspired Therefore this is equal to expire One knows why Yes, my ass one minus two is lost in what people do is is so far this sequence too. It's just what minus two is Why yes, blessed why yes equal Yes, well is squad last four plus one And this is people Places fire minus two s plus Why do iss plus is inspire last for it's just No, I mean Oh yes, he is one Is this fire food into guess Inspire. Minus two s last 10. Yes. What? What? It is far. My last two is last 10. So, for the way we have my tea, this is equal lab lesson worse off one upon it is filing minus two s plus 10. Plus our is into one of our It is fire. Earnest. Yes. Last 10. So it is equally do let Les Waas 01 over is my one All this fire. So it is last three. Oh, these. This is his minutes one correspond. Let's see a smile. Last lesson is all is Squire. Fast food into one over is minus. One will aspire. Nine. So this is written has one about you. The bar teen sign three D Last one of on a tree. Teoh sounds shady and way we have. Why have to one of our key into your body sign treaty glass You need got to d minus sound. Three Time did so the city's Inbar signed signed treaty class deficient utility costs to into T minus tower. You the about town sign down. So this is

Okay, so let's go ahead and just multiply these two factors together before trying to find the value. This integral. So two V times three V is six ft squared and then minus two V plus 15 V in minus five. This is multiplied by T V. And then the last thing that we need to do is just split this up into the integral of the first term minus the integral. The second term plus the integral, the third term minus the integral of the fourth term. Um due to the properties of inter girls were allowed to do this. And then what also we want to do is take out the constants in each of these terms and multiply it by the derivative of just RV variables. Sorry, the integral of just are repairable. So we're gonna have six Times the integral from 0 to 4 of the square D V And then -2 times the integral of VDV And then plus 15. And actually I'm just gonna combine these two like terms, I don't know I didn't do that before but I'm negative Tv plus 15 V is 13. VCR gonna plus 13 times the goal of the T V. And then -5 times the integral of just TV. So for these integral where we have a movie term, what we want to do is kind of the opposite of the power rule for derivatives. And so what we do is we add one to the power and then we divide by that power so here we would have we cubed Divided by three and then multiplied by six. And then we have plus 13 times here again we add one to the power steering V squared and then we divide by that power. And the one place that this technique doesn't work is for V. Two negative first power one divided by V. Since um the derivative of the natural log is one divided by V. So the anti derivative of one divided by V. It's going to be the natural log of E. Um But we don't have to deal with that here. That's just kind of a side note. And then for this last one we're going to have -5 times the integral of Devi which is just be We're looking from 0 to 4. So for this first one I'm six divided by three is to sort of two times to be cubed plus 13 V squared divided by two and then minus five V. From 0 to 4. So in V is equal to zero. All of these terms are equal to zero. So we can kind of just ignore that and just look at when V is equal to four. So we're gonna have V cubed for cube to 64 times two is 128. And then we're gonna have 16 times 13 divided by two which is the same as eight times 13. And let me just plug that into my calculator eight times 13. It's equal to 104 and then we have -5 times four would be -20. So 104 plus 128 is going to be 232 And they were gonna have -20, So this is equal to 212.

All right, so our problem here is why to the full problem when I was four or five equals zero, we're doing that Y zero which is one, why prime is equal zero, Y double prime zero equals negative two And white triple prime, zero equals zero. So we're gonna start this off by just taking a little loss transform each term sort of have S to the 4th, that's the boss transform of. Why minus s cube nope execute why zero? My ass squared white from zero minus S. Why double prime zero, where is s why triple four hours ago, All that -4 Little Postures from Fly Equals zero. So I'm gonna go through and look at the terms above and we have two terms why prime zero and what programs really? Both people Zero, this is going to go zero, I'm just gonna go zero. So here we have wives vehicles one, so you know this whole term is going to go to s cube. And over here we have a negative too. So you know this whole term is going to go to a plus to s It's not rewrite this after having simplify it a little bit. So I passed into the 4th plus stress from of y uh minus s cube plus to S Maya's for it's a loss transfer of why equals zero. Now we're going to separate out the terms that we do most times one of the austrians from y After out of S to the 4th, last four is all equal to s cube minus to us. Okay uh so now we're going to um get the transformed by itself, we're gonna have uh applause transform of why is equal to s cube Maya's to s over As to the 4th -4. Um So now we're gonna try to cancel some stuff out and get this to a form that we can work with so we're gonna factor and s out of the top, see how that goes S squared minus two. So on the bottom we can see that this as the fourth last four is a product of S squared minus four. I'm sorry escort advice to yeah times S squared plus two. Especially these to cancel out and we're left with um S over S. Squared plus. Alright so now where we go from here is um we're going to try to see if this imaginary of plus transfer. Let me know and it looks a lot like the little glass transform of a coastline function. Okay um But this needs to be a square. So we're gonna write like this S over S square plus Square root of two squares scenario, why is equal to the inverse laplace transform of S. Over S squared plus Square. Just two squared. So why is it equal to co sign A Squared of two T. There? It is

Okay. This problem, we're gonna integrate with respect to you first. So either we're gonna have to raise this'll this by no meal to the fourth power, or we're gonna have to do a substitution for that, you plus b squared. So I'm gonna say w is you plus B square. So D w is Do you Okay? So now we have 01 Uh, I forgot this part. Um, if U equals zero than w equals zero plus b squared. So b squared if U equals one than w equals one plus v squared. So now we have the integral B squared to one plus b squared W to the fourth d w and then I still have Don't forget this V here in this devi. So I put vdv. You know what? I don't really like putting that be there. I'm gonna put it over here where it belongs. Right there. So now I was 01 v and I'm gonna integrate this and I get W to the fifth over five from V square to one plus B squared D V. So I have 1/5 01 one plus V squared to the fifth, minus B squared to the fifth and then this V V D v? Yeah. Okay, so now yeah, I have 1/5 01 one plus V squared to the fifth. I'm gonna go ahead and make this into two intervals. So here comes the Vdv Vdv, minus 1/5 in a girl 01 V squared to the fifth, which is V to the 10th. And then B D. V. Okay. I had to give it to both of them. And that's what the square brackets did for me. Can I'm gonna have to make a substitution again. I'm gonna let w equal one plus b squared. Then d w equals to V devi. So I needed to hear which gonna put a one half out in the front here? Yes. If V equals zero w equals one. If V equals one, w equals one plus one square. So too So now I have wanted tint 1 to 2 w to the fifth, D w, minus 1/5 01 B to the 10th times V so v to the 11th d v. So notice on the last one. I didn't change the in points because I did not use uh, you substitution on that 12 totally separate things going on here now. So I have 1/10 w to the sixth over six from 1 to 2. 1/60 to to the sixth, minus one to the sixth. To to the 6248 16 30 64. Okay, so now I have 1/60 64 minus one, which is 63. So 63/60. And then let's see what happens on the other piece of it. And then we'll decide if we need to do some kind of simplifying. So we have here minus 1/5 feet to the 12/12 from 01 So minus 1/60. One to the 12th, minus zero to the 12th. So minus 1/60. And so in the end, minus 1/60 we get 60 to over 60 or 31/30.


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