show-that-each-function-is-a-solution-of-the-given-initial-value-problem-differential-equation-ypr-2

Question

Show that each function is a solution of the given initial value problem.
Differential equation
$$y^{\prime}=e^{-x^{2}}-2 x y$$
Initial equation
$$y(2)=0$$
Solution candidate
$$y=(x-2) e^{-x^{2}}$$

Bobby Barnes · Accepted Answer

I want to show that this is a solution or our initial value problem. So let&#x27;s just go ahead and plug everything in first, see what our differential equation gives us for why Prime. So remember this initial condition is saying when X is equal to two. Why is equal to zero? So let&#x27;s just plug those in. So we get E to the negative so they&#x27;ll just be negative or and then minus two times two times zero. So that&#x27;s just gonna be e to the negative four. When we plug in to now, let&#x27;s go ahead and take our derivative of what are possible solution ists of why prime is going to be equal to Well, let&#x27;s use the product rule. So first derivative of X minus two should just be one. So be it e to the negative x squared. And then we&#x27;re going to add this, too. The derivative of E to negative X. So it&#x27;s X plus or minus two. I&#x27;m gonna race that sign. There is will of minus two because now when we take the derivative E to the negative X squared, that is going to be negative. Two x e to the negative X square because we would use power arena that power and chain roll for this. Well, general first, then power rule. And now, if we were to go ahead and plug in to into this So I&#x27;m not gonna bother d&#x27;oh simplifying that or anything, really, we&#x27;re going to get my prime is equal to So we&#x27;re gonna plug in to now, so X is equal to And actually, I should probably write this out as why prime up to knows it too. B e to the negative or minus two times two times, two times two need to the negative. Or now that there is zero. So this whole term is zero. So you just get e to the negative. So when we checked or our initial value problem, these both gave us the same solution. So it does look like that is a solution for them.

Bobby Barnes · Answer

we want to check to see if this is a solution for our initial value problem. All right, So first, remember, what this is really saying is that when X is equal to E, why is equal to eat? Let&#x27;s just plug everything into our original and then salt, or why pride? So we&#x27;re gonna get E square. Why? Prime is equal to e times E, which is e squared. And then, um, we just get eastward there. So on the right hand side, though, cancel we get zero. And if we divide by B squared on each side, that&#x27;s gonna give us that why Prime is equal to zero. All right, now what we&#x27;re going to do next is take our solution candidate and take the derivative of them. So, um, we could use quotient rule to do this, but I&#x27;m just gonna use product rule. So why Prime is equal to so first taking the derivative of X. That should be erection of using question rule is better. If it was flipped, it would have probably been better to use problems. Eso question rule is low D Hi. Minus hi de lo All over the square of what is below, so that&#x27;s gonna be our derivative. Now we want to check. Why prime a G. To see if he gives us the same value up here of zero. So what&#x27;s so first? This is going to be just one. Now we&#x27;d have natural log e minus one over natural lock squared of E and natural Log of E is one. So we end up with one minus one, which is 0/1, which is zero. So since we get the same values for Y prime, that means this is a solution for our initial value problem.

Bobby Barnes · Answer

we want to show that this initial value problem is a solution for our differential. Equus. Well, let&#x27;s first go to fear. What? Why prime should be, Because then at that point, we just start plugging everything in. So they have why prime is equal to. So they take the derivative that we&#x27;re gonna need to use, uh, product rule to start. So it would be negative. He to the negative action. Let me screw this over so we don&#x27;t run out of room. So why primes, Eagleton? Negative e to the negative x tangent inverse of to heed the X. And then plus, now we take the derivative of tangent in verse, and that should be so first. That&#x27;s right. This so it&#x27;s gonna be one over. That&#x27;s for Marie. You 1/1 plus u squared. So that should be won over. So war he to the two X and then we need to apply chain rule. So we take the derivative of our inside function, which is going to be to e to the X right now. Go Z to the X&#x27;s cancel out there. So that&#x27;s nice. Now what we want to do is use our initial value here and just start plugging everything. Get so let&#x27;s local, the Senate and solve for y prime and see what we get. And then we can plug that value into our equation down there to see if we get the same thing. So this is gonna be why Prime is equal to to over one plus for he to the two x minus. Why now we&#x27;re told that win. So this is X is equal to negative Natural log of two. Our output. Why should be pie? Don&#x27;t Let&#x27;s go ahead and plug that it. So we&#x27;re gonna have to over one plus four. No, he to the negative natural log, too. Well, we can pull that negative to power inside, and I should be e to the natural log of one board. So now the e and natural log accounts out with each other and actually kind of keep on going, And then why should be high half now? Like I was saying, before the those counts out that we have one plus 1/4 which is five force to over by force minus pi uh, which should give us 8/5 minus pi out. So This is what we want to show why Prime is equal to. So now let&#x27;s go ahead and plug this in the white prime. So why Prime is equal to So we want to plug in negative natural auger, too into here for all the exes. So let&#x27;s see. So that will give us negative too Tangent in verse of now e to the negative selections to be 1/2. So we should have Tanja Inverse. Um, one plus then we have two times, Let&#x27;s see one plus, Actually, that&#x27;s the same thing that we had before. Or if we look up there so I should just be the or or 1/4 right here and now let&#x27;s think so. Tangent inverse of one should be pie fourth, but we multiply that by negative. Tues. That&#x27;s negative. Pie half. And then over here we would get the eight equals, but waas plus eight over. So that is we get why prime at negative. Natural too is equal to what we got when we solved or why Prime originally. So it looks like it does check out

Bobby Barnes · Answer

we want to show that this is a solution of our initial value problem. Well, let&#x27;s first just plug in what we have here or our initial value toe. See what we get. So what this is saying is that X is equal to pi half and why is equal to zero? So plugging those into here, we&#x27;re going to get pie, huh? Why? Prime plus zero is equal to negative sign of pie half and sign up. I half is one. So we could go ahead and solve for y prime will be. Why primacy to where we&#x27;d multiply each side by two pies That should be negative to over pie. So Well, we want to check is now or our solution candidate will take the derivative of this and then plug in pie half to see if we get the same solution on. So let&#x27;s see what? Why Prime is so we can use he quotable for this, and I shall bury it. Let&#x27;s use the product rule. So we&#x27;re going to first take the derivative of co side, which is gonna be negative side of X over X and then we&#x27;re gonna subtract again. So it a co sign X over X squared. So that&#x27;s gonna be the derivative. So now why? Prime of pie half. Well, let&#x27;s see what that gives is gonna be negative side. Oh, hi. Huh? Over high half, minus co. Sign of pie half all over. Hi. Have squared. Well, co sign up. I have zero. So this term drops off and then co sign a pie. Half we already said is one and then reciprocating that we should get negative to over high. So this is the same value as what we got by just plugging everything in. So it looks like that is a solution with that initial about you.

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.

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.

Problem

Show that each function is a solution of the give…

Problem 6 Easy Difficulty

Show that each function is a solution of the given initial value problem.
Differential equation
$$y^{\prime}=e^{-x^{2}}-2 x y$$
Initial equation
$$y(2)=0$$
Solution candidate
$$y=(x-2) e^{-x^{2}}$$

Answer

See video for steps.

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals