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Solve the initial value Problem for y as function of x 29) - Vx? _ =Xx>9V(1S) = 0 aX Nx2 _ 81 V3 A)y =Vx2 _ 81 B)y =C)y=Vx2 _ 81 - 9N3D)y =NVx2 _ 81...

Question

Solve the initial value Problem for y as function of x 29) - Vx? _ =Xx>9V(1S) = 0 aX Nx2 _ 81 V3 A)y =Vx2 _ 81 B)y =C)y=Vx2 _ 81 - 9N3D)y =NVx2 _ 81

Solve the initial value Problem for y as function of x 29) - Vx? _ =Xx>9V(1S) = 0 aX Nx2 _ 81 V3 A)y = Vx2 _ 81 B)y = C)y=Vx2 _ 81 - 9N3 D)y =NVx2 _ 81



Answers

Solve the given initial-value problem: $$y^{\prime \prime}-y=9 x e^{2 x}, y(0)=0, y^{\prime}(0)=7$$.

No problem. Twenty axe. But if I do, I already axe it Acts square into this reacts and we got this. Why off terror you colas eight overnight Still first we use separate world was to solve it for Why not here? We want first we can divide X so we have d'lvoire and more idea. Suppose I d close x But if I eat with three acts and we called thiss t X in the integral, uh, both side we have lie equals the integral of this part way Want to use integral by pars. Now, remember, remember, integrate by pars inte gra off you. We pry equals u b minus integral, you prime three, is that correct? Yes, that's correct. So our u it's acts here. We pride we prime is into this reacts. So the is he too. This reacts over story because if you take a directive, you God Okay, we prime. So you see it? Act smart. Applied to this. Reacts over three. Subtract in square off you prime is one. We is eaten. This reacts over three t X Now we got e to the three acts over night back act three actually three miners that the attacks but equally three acts over three subtract e to This reacts overnight plus a constancy. And we have this wise era about there. It was a overnight So we're plugging, actually past era we got his party zero, we have zero here. Elective you two zero ni plus c equals eight overnight. This's negative. Want overnight So see, she is just one hands Our function is why equals acts like by Ito. This reacts over Sorry, but us Each of us reacts over and I plus what so important saying is you have to remember how to into your body parts in this problem. So this is number twenty.

If I want to find the particular solution here than I want to get the general and then fill in the order Param given to the console for C and that's gonna give my final answer. So before I can integrate and want to go ahead and move over this X, which means I have X eat of the three X and then also I want to separate those variables so I would move the D X over to the other side. So now it's ready to integrate. And on the left we get why it's like there's no one here and then on the right side it's gonna be integration by parts. So you is X Devi is e to the three x dx. So do you the driven of there be one D X and V. The anti derivative here would be 1/3 either the three X And as I do this, I kind of did it like a like a reverse chain rule in my mind, just kind of quickly. And I can pretty easily take this derivative and double check I get back to the same place. Has either the three X stays the same times that driven of the exponents times three, which I cancelled with that 1/3 being in front. So now when I fill in that integration by parts ideo UV so 1/3 x e to the three x minus the integral of VDO 1/3 e to the three x dx. The nice thing is, this 1/3 is really just a killer Fish and I could move out front and I already know the anti derivative eat of three X because I did it for Devi so I could just rewrite that last in a rural there already with that anti derivative I know. Call this 1/9 heat of the three X And then there's also a plus c at the end because we haven't indefinite integral here. So now I want to use the ordered Param given from the original set up whatever that might be in this case, zero and eight nines and fill those in in order to figure out what C equals. So why is 89 X is zero? So then, because x zero, this whole first term drops on the second term, that zero there means it's either the zero, which is one, and that just may leave that minus one night there. So if I add that over 1/9 plus eight nine is nine nines were one is the C value. So then the final answer would be why equals 1/3 X each of the three x minus. One night he did three x plus one.

Okay, we're giving the hole in the difference. Your equation and why, Phil, it's one. My prime. I'm only five were asked by the initial They're all your problems. Soft bread. So let's change this into a characteristic equation. And in salt for Amanda, that's limb that minus nine, squared it with a rock star. General solution becomes Why musical To see one need to the nine x c two ex, since we have a repeat need to the Nynex. Okay, let's take the derivative. Who said y prime? Is it good to nine c 18 or a nine x plus nine c two x e tonight, X plus C two seats in the Nynex. Alright, let's plug in what we know more initial values So we know that why go? This is equal to see you want you to tell which one will see to it that you don't actually see two times. So that gives me though, which is equal to why there was one. So we have that. C one is. It was one glass of white time of job that is equal to nine. C one. He too good Carol, That's one plus 92 I'm sterile so that one cancels and to each other Sarah, which is one. So this is equal to life. Can't we know that C one is equal to one? So nine plus C two is able to find This gives me see two people to make it in four. Okay, now let's put that back into a general solution. So we get why is equal to C one, which is one e Gina, my ex plus X where plus or minus four ex each of the nine x.

Number 28 for a while When it goes Boylan, why prime of one equals zero. So then we get r squared plus six R plus nine equals zero articles. *** three. Repeated twice. Before we get wise, you go to see one even I get three z plus c two e to the negative three z and therefore, why is equal to see one over X Q plus c two Ln X over X cubed Why Prime is gonna *** three c one over x to the fourth plus C to one minus three Ln x So the fourth. Using the initial conditions, we find that C one equals one and C two equals three. Therefore, wise you goto one over x cubed plus three Ln x over x


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