5

Points) Find the particular solution of the differential equation dy +ycos(w) = 3 cos(x) dxsatisfying the initial condition y(0) = 5. Answer: y=...

Question

Points) Find the particular solution of the differential equation dy +ycos(w) = 3 cos(x) dxsatisfying the initial condition y(0) = 5. Answer: y=

points) Find the particular solution of the differential equation dy +ycos(w) = 3 cos(x) dx satisfying the initial condition y(0) = 5. Answer: y=



Answers

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-3 y, y(0)=5$$

Problem. Three of section 7.1 asked for us to solve defying differential equation. So the first thing that we're going to do to solve this is we're going to rewrite wide prime, as do I d. X and doing so we're going to separate the variables. So we're going to multiply both sides by D X, and then we're going to divide both sides by why so that leaves us with de y over. Why is equal to negative three d X Now we're going to integrate both sides, leaving us with Ellen of the absolute value of y is equal to negative three x plus c. Now we're going to get rid of the Ln of why. And we need to put Basie on everything to do this. So and this this comes out to be why is equal to e to the negative three x times e to the C? No. The next thing that we can do is we can rewrite this as some very well A. So our equation now looks like this. Why is equal to a times E to the negative three X now what we're going to do is we're gonna go back to the initial condition and satisfy it. So we're setting. Why go to five and we're solving for a by setting X equal to zero and why go to five? So we get five is equal to eight times E to the negative three. I'm zero, as we know. Negative three times 00 And anything to the zeroth powers one. So that leaves us with a is equal to five. Meaning our final solution looks like this wise equal to five times e to the negative three x.

This question asked us to find the solution of the differential equation that satisfies the given initial condition we have D y over D backs is axe sign acts divide by. Why Now we know we we have to multiply both sides by why, in order to get all the y stuff or y terms on the left hand side, and then we know we can multiply both sides by D. X in order to get the ex terms on the right hand side like this. Not integrate. We know why becomes why squared over to us. We increase the exponent bar one divide by the new exponents and on the right hand side, we have negative acts Coastline X, plus sign, X pussy or constant. Now remember, we want this just in terms of singular Why? Which means we have to multiply both sides by two and then take the square root to get rid of the square root and the 1/2 as wth e coefficient we end up with Why is poster minus square root of negative two acts co sign acts plus to sign experts See, now we know they've given us an initial value of y of zeros. Negative. Want we can use this plug in wise negative one x zero we're solving for C. Don't forget. Once we saw first, see, we're gonna be plugging it back into the original equation. This simply means that seat is one. Because we know we're looking at simply negative one is negative square root of sea. We disregard the positive solution because that wouldn't logically sense. It only works if it's negative. Squirt of C two c is one now. Like I said, we're plugging the spot into the equation. We'd figured out why is negative square root of negative two acts co sign X plus to sign acts. The only difference is we're just adding our seed, which is plus one.

Today we're going to solve from the number 17 here The given equation is valued eyes In due course, the square it's plus y minus one equals zero. So why of zero? It was fighting. So why you guys plus one by caused square X in the white equals one by cause the square x here p off Mexico's one because square it's cure fixes equal to one day called square X So integrating factor recalls he integral six square x the X which is equal to us to Panix Why in do it is too planets it was integral into the power Panix in Duke six correx the X right into into the power. Then it's, you know, equals into the power on its plus c. So eat the power Panix into white in tow. The par minus 10 x This one lets see u to the power minus panics. All right, so boundary condition is gonna like y of zero equals five so we can find constancy value for file because one plus see it, the bird minus 10 0 so five equals one plus e in the one see because five minus one, which is equal to four. So Y equals one plus four. Developed by into the power dynamics, which is the particular solution. Thank you.

Right. So what makes this question easier is that it's already basically separated. Everything in terms of X is on one side. Everything in terms of wise on the other side. So to do this so we need to do is first of all when you see why prime for a differential equation you need to solve, it might be it's probably helpful to rewrite that um as dy dx and then now what we can do is just rearrange this. So we have a dx on the other side. Um you can think of it as like multiplying by dx both sides. So basically we get X. Coast X dx is equal to uh to I plus E to the three Y. Um Dy and as you probably would guess we integrate now both sides and uh let's see it for this first side. The left side we'll do integration by parts and you should know how to do this. Basically we'd get X sign x minus the integral of of cynics mm. And uh on the right side we would get let's see yeah integral of two Y. That's going to be y squared. And uh each of the three y. That's going to be 1/3 of each of the three Y. Mhm. Um And we can just put an arbitrary constant let's see here And now um on the left side we get them I'm not gonna keep putting these equal signs on the left side but yeah basically that integral of syntax that's going to be just well nothing negative. That integral. That's gonna kill his co sex. So we get this and the left side can stay the same. Uh huh. Now this is an initial value question. So to finish this off we need to actually find that value. See using um Using the value given which is why zero is equal to zero. So to do that. Yeah we'll put in zero because that's the X. Value zero sign zero plus coast of zero is equal to. Um uh And then the Y. Value 00 squared plus one third of E. +23 times zero is zero. And then plus C. And we're basically you're gonna now solve for C. uh 070 is zero. This is one zero square 20 Each of the zero is 1. Okay so um when we subtract that so we're left with basically one equals one third Plus c. subtract 1/3 from both sides and c. is equal to 2/3. So our final answer is going to be what we got here. Um All right. Um But in in place of uh The C. We can put in 2/3. Yeah Well sorry that that's the that was the previous step where we had integrated the that part yet. So I'm just gonna put that back but it's going to be this this was our final part where we didn't know what she was and uh now we do know what he is. It's uh two thirds. Mhm. But that's not it. Also we want to we always want to try and get as close to isolating why as possible. Um And we want y to be on the left side preferably. Um So we won't be able to completely isolate why explicitly, but we'll do our best and basically when we do that we're gonna flip the sides, getting that Y squared plus one third of E. To the Y. Three Y. Is equal to X. Sine X plus coast X. And I'm going to subtract that two thirds from both sides so that On the left side we just have things in terms of why and we get -2/3 and this is going to be our final answer. Mhm. Yeah. Mhm. What


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