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Point) The differential equationdy dxy? + 10y + 16 cos(c)_ 4y + 26has an implicit general solution of the form F(c,y) K, where K is an arbitrary constant:In fact; b...

Question

Point) The differential equationdy dxy? + 10y + 16 cos(c)_ 4y + 26has an implicit general solution of the form F(c,y) K, where K is an arbitrary constant:In fact; because the differential equation is separable, we can define the solution curve implicitly by a function in the formF(T,y) = G() + Hly) = K.Find such a solution and then give the related functions requested.F(z,y) = G() + H(y)

point) The differential equation dy dx y? + 10y + 16 cos(c)_ 4y + 26 has an implicit general solution of the form F(c,y) K, where K is an arbitrary constant: In fact; because the differential equation is separable, we can define the solution curve implicitly by a function in the form F(T,y) = G() + Hly) = K. Find such a solution and then give the related functions requested. F(z,y) = G() + H(y)



Answers

Think About It is known that $y=e^{k t}$ is a solution of the differential equation $y^{\prime \prime}-16 y=0 .$ Find the values of $k .$

So for this question, we're going to take the derivative of y Z will be e to the power of Katie twice in order to find why Double Prime is equal to K squared to the power off Katie, I remembered the derivative of Kinky is case since Cave the constant. The first primitive is why prime is equal to K E to the power Katie. So the second derivative is wisely with Kate types Cages. Case square into the power of Katie, plugging it into the differential equation. We're going to get K squared e to the power off Katie minus 60 and e to the power off. Katie is equal to zero on in 16 each other. Power of Katie to both sides. We get k squared to the power Kate. Easy because 16 me to the power of Katie or dividing both sides by each of the power of Katie. We get Kenya's equal case. Where is equal to 16 or K is equal to plus or minus four

Okay for this problem. Let's first write it in differential operator form. So that's maybe due to the fourth minus 16 of why is equal to zero. So then now we can find the auxiliary creation P of our is one of equal to our to the fourth minus 16 this equal to zero. So this is actually a difference of squares where this, um, is going to be so r squared squared minus four squared. So then we're gonna rewrite this as so r squared plus four and then r squared minus four is equal to is equal to zero like that. So here we get our is equal to ah, plus or minus two I and then here we'll get ours. He will do plus or minus two. So our for, um okay, our general solution this one we go to first see one, and then a is zero. So either the zero is just won't be one, so they're gonna have sign of BB is too. So sign of two X and then plus C two cosigned of two X and the now plus and then we'll have our real solutions. So c three heats of the two x and then plus C for E to the negative two x So this is our general solution here.

Okay to start solving this offensive question, let's first write it in differential operator form. So that's gonna be D to the fourth minus eight d squared. Plus 16 of why is he with a zero? So then our corresponding on So the recreation p of our is going to be able to our to the fourth minus eight r squared plus 16 and we're gonna set that equal to zero. So we're going to have, um, a difference, Or this is kind of a quadratic equation here. If we instead we had r squared and are, um so let's see if we can factor this. So 16 the factors of 16 uh, that we need to find factors of 16 that add up to eat. So that includes one in 16 which does not add up to eight. Then it's going to be to an eight again. Also does not add up to eight. Let's try for and four. Okay, so four and four does add up to eight. So we can do is factor this and we have our squared inside. Instead of our R squared minus four squared, it's going to be equal to zero. So there now, since we have R squared minus four squared in this square causes that anything we get here is gonna have multiplicity too. So we have our squared is equal to four. And then now we're gonna have our is able to plus or minus two. And then again, multiplicity too, though. So our general solution, Why have X is gonna be C one e to the two X first and then R multiplicity to her that we just want to buy an Exxon front X E to the two X and then plus C three, then e to the negative two x and then plus are multiplicity. Two of that is gonna be plus C four x e to the negative two x So this is going to be the general solution of our problem here.

Okay this is the differential equation that were given and uh so I'm going to let U. Equal to X. Plus one. So then D. U. Is going to be to you D. X. And therefore D. X. Is going to be one half D. You. So D Y. Is going to be U. To the fourth power times dx. Which is one half D. You therefore why is going to be U to the fifth power over five times one half mm Plus C. Now I said that you was two X plus one so it's going to be two X plus one to the fifth power over 10 plus C. Now we know that Y equals six. When X equals 16 equals two plus one to the fifth. Power over 10 plus. See six equals three to the fifth power over 10. Oh my goodness. What is that three to the fifth power? Which is 243? Oh no X zero X zero X. Is not one X zero. So that would be zero plus one. That would be one to the fifth power over 10. So 1/10. Okay. Six minus 1/10. That would be 59 10th. So where were we right here, Two X plus one to the fifth Power over 10 plus 59 over 10. That's it, Yeah.


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