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Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.$$y(x)=e^{-x} \sin 2 x, \quad y^{\prime \prime}+2 y^{\prime}+5 y=0$$.

Solution is defined on interval $(-\infty, \infty)$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 2

Basic Ideas and Terminology

Differential Equations

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in this problem, we are given a differential equation over here and some why of X, and we want to decide if this is a solution to our differential equation. To do this first, we need to make sure that why of X is at least twice differential. Because this differential equation has Why double prime in it and it is, And we can go ahead and find those so the derivative of y vex. Here we have to use the product roll. So let's take the derivative of the first term e to the minus. X times signed two x, and we will add it. Teoh. The first term remains the same times the derivative of second term, just going to be to co sign of two X. Then we can go and find why Double Prime of X, where will need to do the product rule again and was do the same thing. We'll start with this term. Differentiating it. We get e to the minus. X times signed two x plus minus E to the negative X Times to co sign two X, and then we have to do the product role in this term. So plus uh, minus E to the negative. X times to co sign two X plus. We'll leave the first term of the same e to the minus. X times derivative of this guy's gonna be my thrill before Negative sign two x. All right, then we want to take. Now we have this. Why prime my double crime and we want to plug them in to the white double prime my prime. Well, put it this way of accent of the white here, and make sure we get zero. So let's go ahead and do this down here. So why double prime is all of this stuff, so we will go ahead and rewrite it e to the minus. X times sign two x minus to e minus x co sign two x Fix that to a little bit. Yeah, um, minus Teoh e to the minus x co sign of two X plus, Actually minus for E to the minus. X sign two X. Okay. So that's why double Prime. I just rewrote simplified parts of it and we're rewriting this equation. So that was why double prime plus two white crime, um, negative e to the minus. X times sign to eggs plus e plus simplifying to e to the minus X crow Sign of two x So that Waas why prime? And then finally we're going to add plus five why? And instead of why of X we're gonna right, But we have over here e to the minus X sign two X Now we're gonna help we get some cancellations here, so let's change colors. Here we have eat the minus X sign two X. Well, that cancels with one of these Negative each the minus X signed two X So instead of two here, we, uh, here what's actually we distribute this two out? So it's here, and it's over here. So this whole thing becomes negative to be to the minus X sign Teoh X plus for E to the minus x co sign two X. All right, So in that case, yes. With at this term right here, cancels with just one of these negative e to the minus X into X so we can get to this too. It's now one. Uh, we're gonna wondering There Okay, uh, minus to eat of the minus. X co signed two x. Well, that cancels with two of these positive Teoh e to the minus X got assigned to X. So we get this. Four becomes eight to and this goes way right now we have minus two e to the minus. X coastline two X, that is the same thing is one. Before we get rid of that cancels with these remaining to here and this term goes away. And here we have minus for each of the minus X signed two x and we have five of those here and, um, one of negative one of those left here. So here, in total, we have minus four Eva, minus x side of two X and two minus one e to the minus x sign of two X. So if we combine this term in this term, we just get five started Negative five. You know, the minus X times sign two X and that is going to cancel with our five of the X in the minus X sign of two X and we get zero Then the last thing we need to dio is just to make sure that we know what interval this is true on. Well, if we look at our right of the X, even the minus x X could be any real number and sign of two X X also could be any real number inside of sign. So sign and Costa. And don't put any sort of stipulations on what X can be and same thing with eat with X. So we're good and we know that ex convict be in any of the real numbers or X we is in negative infinity, too infinity.

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