show-that-each-function-yfx-is-a-solution-of-the-accompanying-differential-equation-yprimey2-a-y-f-2

Question

Show that each function $y=f(x)$ is a solution of the accompanying differential equation.
$$y^{\prime}=y^{2}$$
a. $y=-\frac{1}{x}$
b. $y=-\frac{1}{x+3}$
c. $y=-\frac{1}{x+C}$

Bobby Barnes · Accepted Answer

We want to show that these three are solutions of our differential equation, so we could just take the derivative of each of those three functions, plug a man and go from there. But one thing that may help us cut down on a little bit of time is if we solve for C first and then use that to answer a and be released to show that it holds current. So let&#x27;s start with C. So let&#x27;s see. What? Why prime ISS or C? So we would write so negative deep I d. X of X plus seat raise the negative first so we would use power rule to take this derivative move the to outfront. Subtract one off of it. The negatives cancel out so that it&#x27;s just going to be X plus C rays to the negative second, which we can reciprocate as one over X plus e to square. All right, so that&#x27;s what Why pry Ms. Now let&#x27;s see. What why squared is so why squared is Nega Do one x will see squared, which is just going to be so the negative goes away since we&#x27;re squaring it and it speaks one over x plus e square. So why squared is equal to y fronts? Coursey, Right? So see checks out now for parts A and be actually. So for part, A A is the case. So it&#x27;s heart. See when C is equal to zero so that one will check out as well. And B is part see when C is equal to three. So by just showing it holds for see first, it&#x27;s Trivoli follows that A and B work as well.

Bobby Barnes · Answer

we want to show that these three our solutions or our differential equation Why is he go to why Prime plus three y is equal to e to the negative X was just check to see if they work out or not. So let&#x27;s first figure out What? Why Prime ms for each of these. So for the 1st 1 there&#x27;s going to be why Prime is just gonna be negative. E to the negative X. Now we can go out and plug everything in to our differential equation right here. So it&#x27;s going to be two times why Primes. Let&#x27;s just multiply this by two and then three times why? And we want to add these two equations together. We&#x27;re gonna get do you buy? You&#x27;re right. It same way they have it too. Why Prime plus three by is equal to three minus to eat the negative X which would just be eaten named Rex so that one checks him. Now what about the 2nd 1? Well, this is going to be why Prime is eager to. So it should have the same derivative as First Minister e to the negative X and now this should be negative rehabs e to the negative x And again we would multiply the top one by and we&#x27;re gonna multiply the bottom one by two. Actually, those two&#x27;s counts out with each other. Now, when we add these equations so on the left side also have the two by price plus three Y, uh, three e to the negative x plus a negative, too. Heat of the night to extract. Still be e to the negative X and then the e to the negative x over to R E to the negative three House X Well, for those will cancel out with each other, so just be there. So that one check so and then lastly, we can find the derivative. Oh, see? So why pry? There&#x27;s going to be equal to e toothy. Negative Thanks. Her negative e to the negative x plus. Now it will be negative. Three, huh? See, he too negative. Three x over, too. And just like before, we can go ahead and multiply our first equation by the re smoke three. But me you could read So 33 and three And the next one we&#x27;re gonna multiply by 22 and two. The twos they&#x27;re cancel out. And then once again, we get to why Prime plus three y is equal to so three e to the negative X minus two e to the negative extra S B e to the negative X and then the last terms here. Those were just cuts out with each other when we add so that one checks out as well. So it looks like all three of those are solutions of our differential equations.

Stephen Hobbs · Answer

to verify this. Um, given equation up top here that the derivative is equal to the original function squared. We&#x27;ll go ahead and take the derivative, but first we&#x27;ll just rewrite each of these as their power. So the first one would be negative x to the negative one, and then we&#x27;re ready to differentiate using the power rule. So why prime would then give us the negative comes in front and it would be one over x squared if you notice. If you were to square this here, it would end up giving you the derivative in the bottom. And so this first one is confirmed, it&#x27;s gonna be a similar process for all the rest of these. The next one is why just to rewrite it. That&#x27;s the same thing as negative X plus three to the power. Negative one. Take the derivative. Why? Prime is the power rule and the change will. And that would give us one over X plus three on it would be, um so the negative two. And then, of course, that would really just do the same thing. Is all of this squared which matches with if the original function here was squared would get the same thing as the derivative. So this one matches with this Ruelas Well, and then, lastly, we have Why is equal to negative quantity X policy? All So the negative one. Then why prime taking the derivative negative negative would be positive. Then it would become toe the negative, too. And that&#x27;s the same thing as X. Let&#x27;s see all of that. Where and that then again is the same thing as this original function. All of this squared is the same thing as the derivative, so each of these is true.

Bobby Barnes · Answer

we want to show that why is a solution of our differential equation here? Well, let&#x27;s go ahead. And Vera, What? Why Prime is going to be to start. So why Prime should equal to Well, this is going to be product rule, so it&#x27;s going to be so I&#x27;m gonna burst right there is X to the negative first accident. Get a first power. Well, this should give us X to the negative second times, negative one, and then are integral. So one x e to the t over tea ct, and then we&#x27;re gonna add so we still have X to the negative one, and then we need to take the derivative of this. Well, that&#x27;s just the fundamental theorem of calculus. So we would just plug in X or tea. So should be e the X over X. And then let&#x27;s simplify this down or make it look a little bit nicer at least. So this should be negative one over expert that we have integral from one two ex of e to the t over t t t. And then this is gonna be plus e to the X over X squared. So this is why I pry and now we want to multiply. Why? Prime by X squared? Let&#x27;s do that. And then notice that we get why prime Times X squared is equal to Bill. Both of those experts cancel out and we just have negative of are integral and plus E to the X. Now we want to multiply. Why buy x so X? Why is going to be well, that just cancels out with the one over X out front here. So we&#x27;re just left with our interest. And now if we were to ad these two equations together adding them, we&#x27;re going to get X squared y prime plus X Y is equal to well, our intervals will counts out since they just have opposite signs of each other and we&#x27;re left with eat the ex so it looks like that is an actual solution.

Stephen Hobbs · Answer

to begin with, we could just square each of these expressions A, B and C to find out what this would give us in the first woman. It would be one over X squared, and the next one it would be one over X plus three squared and then similarly, for the last one would be one over quantity X well, us see square. So we know what each of these squared parts on the right side should be. Now we&#x27;ll go ahead and take the derivative Another way to rewrite that first one. This is the same thing as negative X to the negative one. So when we take the derivative, it&#x27;ll end up being the negative would cancel when it comes to the front. And so it would be X to the negative too. But of course, that&#x27;s actually just the same thing as one over X where so that first one is true. That statement up top is satisfied. The next one, the rest are going to be very similar processes. So negative X plus three. Another way to write one negative one over X plus three is negative baffle quantity to the negative one and then again power rule that negative comes in front. And then that would cancel. And so it would be one over X plus three, all of that square. So again, same thing as what we have here. So the second one checks out, and then the last one, part c Negative X. We&#x27;ll see to the negative one with that negative front, positive X plus C to the power. Negative, too. And again, what&#x27;s the same thing? That&#x27;s that&#x27;s all three leads work.

Stephen Hobbs · Answer

Our goal in this video is to make sure that all of the equations given match with this differential equation. So first off, taking the derivative of the first function of B why prime would be, um e negative e to the negative X substituting that back in to the function on top. We&#x27;d have to times that so we could just go ahead and say negative to B to the negative X and then we still have plus three Why, which is just this equation. So three e to the negative X. Now we want to make sure that this does, in fact, equal even the negative X well based off of the constants in front here. Then, of course, this just gives us one either the negative X or just e to the negative X. So this first one did work and did satisfy the top equation. The next one that will look at is taking the derivative of this function, and we&#x27;ll get negative e to the negative X and then we&#x27;re going tohave minus three have in to the negative three heads X. Now this one is going to be a bit longer, but I think we could fit it in the next step here, so it&#x27;s gonna be two times all of this. Let&#x27;s go ahead and write it out. E negative E to the negative X minus three pounds. The so the negative three hands x all of that now plus three multiplied by the original function, which is in blue there we just got and put it back in blue. So it&#x27;s easier to see e to the negative x plus. So I&#x27;m negative three pounds specs and all of this. We want to be able to simplify and to just get you to the negative X here. So let&#x27;s see what happens. What we&#x27;ve got to e negative to eat of the negative x positive. So these two will combine to e to the negative acts. Let&#x27;s see about these next term&#x27;s here. Well, the twos canceled here from the front and so that would be negative. Three. This second one here would become positive three. So this one positive three that negative three this and so these ones would cancel. So that would just leave us with each of the negative X And then from here we just have a more generalized version of it below. And so let&#x27;s just jump right into it. So we know it&#x27;s going to be two times y prime, which is negative e to the negative X and then minus three on D. C mm to the negative three hands X and all of this plus three. We&#x27;ll go back and say, plus three times the original function that you could just put back in blue here. E the negative ax once. See he the negative three. It&#x27;s, uh, okay. The first two terms negative to either the negative X positive three. Either the negative XP&#x27;s would combine e so the negative X the next term&#x27;s the two would cancel with this, too. So that&#x27;s negative. Three c, etcetera. This one&#x27;s positive. Three. After distributing, see etcetera, positive, negative negative cancels with this positive. So we are in fact left with this expression as well. So all three of these do fit this differential equation here, and all of them satisfy that solution

Problem

Show that each function $y=f(x)$ is a solution of…

Problem 2 Easy Difficulty

Show that each function $y=f(x)$ is a solution of the accompanying differential equation.
$$y^{\prime}=y^{2}$$
a. $y=-\frac{1}{x}$
b. $y=-\frac{1}{x+3}$
c. $y=-\frac{1}{x+C}$

Answer

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University Calculus: Early Transcendentals