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Point)Given u(x,t) = /e" compute:UrX...

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Point)Given u(x,t) = /e" compute:UrX

point) Given u(x,t) = /e" compute: UrX



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Find $d y / d x$ at the point (0,1) using implicit differentiation given that $x y+e^{y}=e.$

Yeah. In this problem we are given the differential equation of a co Which passes through the zero, command zero. This is the differential equation of the girl. That is why prime is equal to M4 x times of silence. So how we do this is we first find the general solution for this differential equation. We then substitute this point that is 000 into the general solution which will help us to get the particular solution. And this, in fact, is there equation of the group? So let's go ahead and find this one. Uh We have this white primary call to a poor ex Synnex. Notice that this white prime is another notation for dy dx. So we can they do this as do you have a dx is equal to X cynics? And then multiply both sides by dx. And when I do this this dx will get cancelled and we will have only divide this site. And on the right side we will have a poor X times of cynics dx. So as we can observe that both sides, we have separated the variable that is on the left side, we have only the white terms and the D way. And on the right side we have all the X terms as well as their dexter which means we can integrate it and then find the solution. So let's do that. Well, I don this integrate this is a question. I'm going to bring it over here. It is D Y a call to people, X cynics, the X. So let's integrate both sides. The integration of the website. That is which is the way is simply why. And the integration of this epoch sign next year's. We are going to do this using the integration by parts minted integration by what's meant that since this is basically a product of two functions we have those departments which is an exponential function and the syntax which is to trigonometry function. So let's integrate this using integration departmental 1st. Let me assume this. I integration as I equal to a poor X. So the knicks dx and in integration by parts we have this formula you D V is equal to U V minus integral video. And the choice of this you actually based on a rule and this rule says uh shortly denoted as I L A T E. Which is basically the ice transport in verse. And this stands for low algorithmic A. Stands for algebra. The stands for technology and Eastern Sport Exponent ship. And we had to observe in this order. So when you look at these two different functions we have exponential that is epo X is an exponential function and Cinemax S. A. Trigonometry function. So when you absorb this in this order I E L A T E. The city comes first which means we have to choose us this syntax So we choose you equal to cynics were I don't like this and then we separate the list of the things uh that is a poor excuse. Once we have selected this cynics as you we have selected this as you and the rest of the things that is a poor X times. Rdx will ride on that as DV this is equal to people X bx. Okay and in this method we have to take the differentiation of you on both sides. This one we have to differentiate as well as the signings and when we do that we'll be getting D. U. Is equal to differentiation also in texas Cossacks. And then we put this bx like this and this side we have this D vehicle to epoch sonics. So we integrate both sides. So integration of devi is just we and integration of people exist E power X. So that's what we got now. We can substitute in this formula. So this basically which means this i is equal to april x syntax D X. This is what we have. And using this formula we have to first multiply you and me. That is sign it's time soft uh politics. We have this you over here multiply with we when we do that we will we will be getting a poor X. Times of cynics and then we have this integration of video. If you look at this formula we have this UV minus integral video so we have to put integral we is e Parex and then there's D'you we substitute for the US well this is study you so therefore it will become Cossacks. Bs Now we still have to integrate this part. This part we have to integrate again. Also again we are going to the integration by parts meant that so uh we have to follow the same rules at least. We have to first uh select the you and then we have to select the rest of the things as DVD. So when we use this rule, these Cossacks we can select this as uh you. This time I'm going to write don't as a human And this DVI'm going to write don't ask the DV one. So basically you will is equal to Cossacks and DVD one is the uh terms uh other than the cost of that is we have the poor x dx and once again disturbing here to take the differentials on both sides which means we'll be getting de you won is equal to differentiation of course access negative sine X dx. And this side we have to integrate Integration of DVD one is we want is equal to integration of the product Z products. So therefore we have this formula you be one minus that Is you and we want minus integral of B one. Do you want we just have to change the notation by substituting the subsequent one for all the variables. So we can substitute in this formula. Remember we are going to substitute for this part since basically we are integrating this part so I'm going to I don't hear so we have I equal to the side we have I equal to I'm writing this to me Parex syntax minus. Let me put a bracket. Uh here I'm applying this um we won formula which means this U uh U N. S. Cossacks and we even is F or X. So therefore this will become a poor X. Call sex and then minus we have this minus or hear integration of V one. Do you want? So we substitute the we want do you want? We want is a poor X and D U one is negative Sine X deployed by dx. So this is what we have. Let's just simplify this. So the side we have I is equal to people X. Cynics minus off we can distribute this negative to both these terms, this will become minus F or X. Call six. Once we distribute this minus we already have to negative, that is too negative ones. So minus and the minus is plus once again plus into minuses minus. So we can put this as minus off a poor X. So the next the X notice that uh we already assumed this term as I as you can see from over here we have in the beginning we have initialized or we have I assume this as a vehicle to a poor exercise next year. So therefore when it comes over here we can replace this as I this could be the center term could be replaced as I. So therefore begetting Perec's cynics minus off epo X core sex minus. Which means if we add I on both sides will be getting two over this side and decide to be having the product Synnex minus the products causes which means manufacture the products. I'll be getting a product sin x minus cassocks. Now I just have to find the value of why So therefore I do it both sides by to which means I'll be getting equal tooth people rex cynics minus of cossacks all the way boy too. Ah let's go and find Hi! So I is basically the integration of people X nx dx. We have actually started off with this one. This part You can mark this as equation one. So in equation one we can substitute the value of so therefore will be getting the solution. That is a general solution is why equal to I. So we can I don't discuss decide as we have white and I said we have just like that is a Perec's cynics minus of Kazaks All divided by two. Then we have to put the constant of integration seat. When you look at the Equation one we have to integrate both sides. So we integrate will be having this constant of integration. So this is basically the general solution of the Cuban differential equation. Since we have to find the equation of the car which passes through the point zero comma zero. We now substitute this point into the situation which will help us to find the value of seats. So we plug in zero comma zero for both X and Y pregnant. This value when we do that we'll be getting zero equal to here. S to the power of zero. Sign zero minus Call zero All the way by two plus C. So let's find the value of C. This is zero epo zero is one, sign zero is zero. So can cancel this and call zero is one. So therefore this will become negative one x 2. Let's see which means we have a sequel to minus R. I'm sorry, zero equal to minus off policy. So this employees equal to half. We take this value of C. And plug it into the general solution. That is into the situation why equal to people X times of sin X minus course takes place, do it with two plus C. So let's do that. We have Y equal to people X sign x minus of call six. All over to plus half. We know multiply both sides by two so that we can remove this fraction. So this will be getting To Y is equal to four x cynics minus of cossacks plus one. We can write down this s when you subtract one on both sides to Y -1 equal to e products claims of sin X minus kazaks. So this is the equation of the girl For the given differential equation and passing through the points recommend zero.

This problem were given any question And where's my secondary toe? Why would suspect X clinics is here? So let's first place exterior and and let's see the value hope it function while we're next. Zero. So we have zero times. Why? Plus Egypt A Y is able to be this term is Europe. So from this we see that when x zero y is equal to one All right now fighting for Sarah to have less differentiate bolts X we have why plus X y plane plus each other. Why times y prime music. Zero firmness. We see that Then why Prime is equal tornado You wanted eggs plus each wine What x zero We found that wise a good one less evil 80 the average of at that point. So why prime would then be the originator of one or were zero plus one and that every negative would over eat? All right, now, using this white prime this equation, we're gonna depreciate it there one more time to find secondary tive will use caution. Trolls are white devil would be equal to the negative off. Why prime that was X plus each other. Why plus y times one Waas eating. Why find more prime divided by X plus each other? Why, it's Kurt or it now, Unless you let why? Double prime secondary Quiet one x zero. That is negative. We found why Prime is a coordinated he so we haven't already. Times 01 e plus one times one waas e conflicto e You added by zero plus c squared that these four cancel out. This would cancel out and we didn't find one double time to be negative off negative walk over the script for the answer is one over the square.

In this problem we will cover the differential of a two variable function. So I have written here in green, the general formula for a differential of a function at a point A. B. And we see that we will have to find the partial derivatives with respect to X and Y. So we will first find the partial derivative with respect to X. Of our function F. And that is basically taking the derivative while holding the variable Y fixed, some taking the derivative of the X term while keeping the white term the same. And that yields us E. To the negative Y. Since the derivative of X is going to be one And our partial derivative at the .10 Is just going to be one. Since if we plug zero in for Y. E. To the zero is just one. Now to find the partial derivative with respect to Y. And here, instead of holding the variable Y fixed, for going to hold the variable X fixed. So we can just move heads to the front while we multiply by the derivative respect to why of each of the negative. Why? And that is going to yield this negative X times E. To the negative Y. And if we were to plug in the .10 into this partial derivatives, we will get negative one because once again each of the zero is just one and you plug in one for X is negative. So now that we have our partial derivatives, we can create our differential. So we have D. F. Is equal to one times the X. Which is just the X. And -1 times dy. Which is just negative deep y. And here is our differential for our function F.

We want to find one that we're trying to disfunction at this point. X zero. Let's find the Y value. Furs executes 20 We just fall into here. We get zero plus either power. Why? It was e. This is really power one. So my wife is one. So this is the pal point that we want to stop in for to find our white crime data and our wind up a crime. So let's differentiate dysfunction. We respect to X on both sides using for that rule, it frees the X differentiate the wine you get white crime Last place, the y you can shoot the X. There's just the one plus either power while just be either about white and then differentiate the y I get white plan equals toe is just a number. When you differentiate, you get zero. So from here we set in Mexico zero wife was the one into here You will get this zero plus one plus even about one white prime because toe zero So my wife find is minus one over he Now let's differentiate. Let me call this equation one. Let's differentiate again. One with respect to X on both sides. What do we get? So over here you will get X again. Why the good time? That's now. Why prime depreciate. It's like that one last no differentiate the why he I get white crime. Plus here I I fist either. Paul, Why differentiate wine prime? I get white double prime Plus now I face my wife. Prime differentiate Either power. Why? I'll just get you to pa y and then white Prime, It was 20 Now let's stop X equals zero. Why? It was toe one and why Prime equals toe minus one over E into here into this partner. So we will have zero plus minus one over e plus. Okay, so there are two of them, so I can just put two here. Plus over here is eat about one. Why the book? Crime Plus now White prime here is minus one over e and I e to the power one. And why prime here is minus one with e equals to zero simplifying you will get why the prime equals toe one over the square


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