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Find the equabon of the Iite that goas through (2, 9) and perpend cular {0 Y =Wrte Ihe equatlon the (orm *-ayeb; Or y -=mx+b.oquolion(Type the equation using Inlege...

Question

Find the equabon of the Iite that goas through (2, 9) and perpend cular {0 Y =Wrte Ihe equatlon the (orm *-ayeb; Or y -=mx+b.oquolion(Type the equation using Inlegers or fractions . Simplily your ansket )

Find the equabon of the Iite that goas through (2, 9) and perpend cular {0 Y = Wrte Ihe equatlon the (orm *-ayeb; Or y -=mx+b. oquolion (Type the equation using Inlegers or fractions . Simplily your ansket )



Answers

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}+9 y=\frac{36}{4-\cos ^{2}(3 x)}$$

Okay let's solve this problem by solving for the modular solution. So I have the R squared was nine equals to zero. We can move the nine to the other side. So have the R squared equals the negative nine or that are equals to this equals to plus or minus the square negative nine. Which can be simplified down plus or -3. I. And so with this we can actually build our homogenous solutions or modular solution in this case. Can we see one co sign three x plus C. Two Sign three acts. We can also take a stab At our guest for the particular solutions. Our guests for the particular solution is going to perform a coastline three X plus Be signed three x. Now note that there are these terms that are matching in the Hamas, a solution and the particular solution. And since we don't want that, what we end up doing is we end up multiplying by a factor of X in the particular solution. And so now we can take the derivative, this particular solution twice. So the first derivative, It's gonna be a co sign three x Plus be signed three x. And the second derivative is going to be negative three A sign three X plus three B. Co Sign three X. All times X. And we're going to take a better out of the skin. So have Y. P. Double prime Equals 2 -3 A. Sign three X plus three B. Co Sign three X. And again, we're going to do the product rule on the right? So have plus or minus three. A. sign three x Plus three b. co sign three x. And then the truth inside what is inside the parentheses is gonna be negative nine A coastline three X. And the other term is going to be minus 90 signed three X. And this all is multiplied by a factor of X. So these two terms right here multiple dialects. And so with this we can actually simplify it down a little bit. And so we'll have that these two terms are here can be condensed and also these two terms right here can be convinced. So when we simplify it we should get negative six A sign three X plus six ft coastline three X minus nine A. X. Co sign three x minus nine B. X. Sign three X. Okay, so now what we want to do is we want to plug this in to original equation that we had. So original equation was Y double prime. So negative six A. Sign three X plus six B. Co sign three x minus nine A. X. Co sign three X minus nine B. X. Sign three X plus nine Y. So in this case we said why was A X. Co sign three X. O plus nine A. X. Co sign three X Plus nine b. x. co sign it's our sign three X. And this is all equals two. Sign three X. Okay. So let's do some simplification before we move on to creating a system of equations here. And so let's notice that this term and this term are the same. So they cancel each other out and also that this term and this term are the same. So they cancel out. And so what we're left with is negative six A. Sign three X Plus six b. co sign three x Equals to sign three x. And so now we can build a system of equations here. So we have the -6 a. Equals to one And that's six B equals to zero. And so with that we can draw that B equals zero And that a equal to negative 1/6. And so now we can plug this into our original particular solution guests or a particular solution guests initially was negative. Will it be negative? 1/6 X. Co sign three X. And with this we can actually build our total solution which is the particular solution plus homogeneous solution. So I have that total solution is negative 16 X. Co sign three X. In this case we said are homogeneous solution? Will see one co sign three X for C. To sign three acts. And so that's our answer.

We have the question off the line and our aims to find the inclination data off the line. We know the relation between slope and inclination off a Lina's M s call to transcend off inclination, and we have the standard form off the light so we can write this slop as if express B y plus Z is equal to zero and slope and make will do minus a by B. So here now line. The slope is minus four by five, so we have slope here. Now we can find the using the bicycle do turning worse off em. But for am greater than told to zero, we have to add a pie extra to get the correct value off Tita for negative values off him. So we have used a second expression that is, the type equals by plus turning worse off minus four by five that is equal to 2.467 Radiant or in degrees. We can convert this two degree and obtain the values when 41 point 34 degrees. This is our inclination

Rolled. So just bear with me. Um, the original equation. We have just gonna get this on the slide so it can refer back to it several times. Is that X squared? Minus three. Route three x y for 65 square plus four y minus three equals zero. Okay, so then the main thing I want to seduce to choose a data to rotate this comic section through so that we can, uh, remember X Y term and be able to to perform our normal operations with like, preferable is ellipses and problems. Okay, that being said, we need Teoh evaluate to find that data by using the formula co tangent of two times. Our data is equal to a minus. C Overbey. I can certainly use some space to solve out what our fate iss so a minus C That's 9 96 over B and B is negative. 3/3. Negative three, Route three. So we're gonna have three over negative. Three rude three Sarko tangent is just gonna be negative. One ever read three. Okay, so containment is adjacent over opposite. So if we were to draw a quick triangle Well, that's going to be one of her room, three sort of reference angle is gonna be 60 degrees. So tooth data, it's gonna be 100 and 20. And so they've, uh, is gonna be equal to 60 degrees. Is this the first part of our question? Now, the next part, we have to actually write out the new equation rotated 60 degrees. So the next time I'm gonna calculate with our new X and Y values will be based off our formula for the rotation. So X is gonna equal Ex prime sometimes co sign of 16 minus one crime time sign of 16 are why we're gonna replace with ex prime times. Sign of 60 plus. Why prime? Because I m 16. Okay, so these values simplify down. Teoh co sign of 60 is 1/2. So we're gonna have ex prime. Both is easily over two minus. And in our sign sixties, route three over twos. That's gonna be in minus for it. Three times like prime came by the same, uh, problems we're gonna have rid three. Thanks, Prime. Plus, why? Prime for two for one value. Okay, so now we're gonna plug these x and y values back in to our original equation. So my credit new side refer back to my first equation, and I'm gonna leave some gaps each time we have an X or A y. Okay, so we have you nine times are X quantity squared minus three. Route three times are X quantity. Times are white quantity. Kind of have plus six times. Or why calling b squared plus four times are right quantity and then minus three equals zero. Okay, I left these big spaces, cause we remember that our X and Y coordinates are going to be plugged in, and those route will be then. Okay, so our X coordinates gonna be X minus. Rude three. Why? We lose the primes for now? I just to kind of make it unless to click. Yes. Okay. That are wise. Will be, um, Route three X. That's line over to for every three x plus. Why? Number two and red three X plus y over to. Okay, so slide four. I'm actually going to start expanding this out. It's a painful process, but we just got to make sure we were careful and we keep track of our right coefficients. Okay, so for our first element. Uh, our denominator squares just four. Ah, eso We can pull out the coefficient of 9/4, okay. And then expanding out our numerator, we're gonna have X square minutes to route three. Next, my plus three y squared. OK, so that's our first term. Our second term. The denominators multiply to give us four, and so are coefficient is just gonna be three. Route three over four and then multiplying. Are enumerators gonna expand it out? So could be a weird expansion. I guess So. We have right three x square. Just kind of foiling. Ah, we have X times. Why? It's just plus X y. We have negative route three wide times, right. Three exits minus three x lie in our final term will be fruit three white times. Blang, which is minus fruit. Three. Why square? Okay, our next term or third term by squaring the two in the denominator becomes a four. So we're gonna have six over for my keep our denominator being formed. I couldn't reduce. That's because our other coefficients had four in the denominator. Hopefully it'll make it easier later if it doesn't. That's OK. I read three X squared is just gonna be three. Thanks. Square we're gonna have plus two, Route three. Thanks. Lie. And then Plus why square can our third term, we could pull the two out, So just be plus two, the times Route three x plus y minus three equals zero. Okay, Um, yeah. So our next slide that we're gonna calculate, um, what are different? Coefficients are so are a coefficient. Is the coefficient in front of our X squared? Get there and find all of our coefficients that have an X squared involved. Okay, so we have a 9/4. We have a route three times. Negative three. Route 3/4. So that's gonna be great. Three times, right? Three is three. So minus 9/4. And so that takes care of this. Takes care of that. One. Takes care of that one. And there are last X squared term is here. So three times six is 18 and we have 18/4. Uh, 18/4 is just nine over to for a bow. Would be nine over to can be value. That's our X Y terms. We're hoping that this goes to zero. Okay, So for our first ex wife term, we have 9/4 times. Negative to root. Three. That's going negative. 18. Route 3/4. Case it for our second lines. That takes care of that one. We're gonna have, uh, the interior of our parentheses Have negative two x lie in the negative. Two times. Negative three. Route three is gonna be positive. Six. Route 3/4. Good. That takes care of those two terms in our last term is too rude. Three times 6/4, which is 12. Route 3/4. And that does. In fact, I'll cancel out to give us zero, which is what we wanted awry. We're doing something right. Okay. Are see value. That's horrible. I square term. We're gonna three times, not ever For which is 27/4. That's our first print. It sees our second wannabe negative ring three times. Negative. Three. Rid three. So it's gonna be positive. 9/4 takes care of our second term in the last term is just gonna be Why Squared Times six over forces six y squared over four and that's 27 plus nine is 36 plus six is 42 said 42/4 is just 21/2. There's our see value care devalues our ex term. We don't have a how we do the next time. Okay, so we've already used all of these three terms, so we're just dealing with our final piece here, and we basically just that's gonna be our d e and F. Um, said the is to root three. Uh, he will be just to and f will be good three. Now, the nice thing about this all equaling zero, we can come up with an equivalent equation without fractions, but it's multiplying everything about two. So we're gonna have our seed eyes. 21 are able use nine our devalue before route three. Sorry. Value before. Right after I be not negative. Six. So that gives us our equation. But I'll go ahead and couldn't decide and write it down. Yes, we're gonna have nine ex prime squared plus 21 wide Crime square plus four. Read three. Ex prime was your wife. Prime minus six equals zero, and that will be our final equation. Rotated the 60 degrees that was needed to get rid of our X Y term. Thank you very much.

We have to find the inclination off a line whose standard equation is given. So we have the relation between slow, pent inclinations and slope a sequel to times and off inclination and from the standard form off the line, which is eggs. Please be right. Plus C equals zero we can find the slope is m equals minus a baby. So here the slop er's minus one by Route three. So Emma's negative value so we can find the inclination teachers pi plus China in verse off AM Where am less than zero. Okay, so here, by place, turning worse off minus one by three, it will give us value off intonation as 2.618 radiance or converting two degrees. It is equal to 1 50 degrees. This is our inclination.


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