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A ? I} " Selber abeo an 5' 5' an correct snowouoine Hulu a repeller differential equation sla { the critical point...

Question

A ? I} " Selber abeo an 5' 5' an correct snowouoine Hulu a repeller differential equation sla { the critical point

A ? I} " Selber abeo an 5' 5' an correct snowouoine Hulu a repeller differential equation sla { the critical point



Answers

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. $$f(t)=\frac{1}{5} t^{5}-a^{4} t$$

That's suspected of river overhauling function to find a critical point. So we'll start by taking care of it over this term. So that's 5/5. So that's one. Actually, there are four minus aged Bar four. Okay, so let's see if we could unpack this so we can practice into X squared, plus a squared times. Um, exports. A Times expert is a and I will set this equal to zero. So this term gives us imaginary numbers. So we see that are critical points are these two terms? So that's one X is equal to negative A. And when it's equal to a

To solve for the initial value. Probably 1st integrate both sides of this differential equation and we will get Z equal to the integral of X cube times Ln of X dx. And in here we have to apply integration by parts we want to let u equal to Ln of X DV is going to be the remaining expression execute the X. And for me we will get differential of U equal to one over X dx. And we is going to be the anti derivative of devi that's equal to x rays to the fourth power over four. So by integration by parts Z is equal to U times v minus. The integral of the times do you? And that's going to give us Z equal to x rays to the fourth power times Ln of X over four minus the integral V which is X rays to the 4th power over four times the U. S. Has one over X dx. That's equal to x rays to the fourth power times Ln of X over four minus one. Fourth times in the role of x rays to the third power dx. And by power rule, that's just x ray to the fourth Power times Ln of X over four minus 1/4 times x rays to the fourth. Far over four. And then plus C what is the same as x rays? To the fourth? Power times Ln of X over four minus x rays to the fourth. Power over 16. And then plus C. Now to find, see we will use the fact that Z of one is equal to five. So if ZF one is equal to five, then we have five equal to one race to the 4th. Power times. Ellen of 1/4 minus one race to the fourth Power over 16 plus C. This gives us C equal to 81/16. Therefore our solution is Z equal to x rays to the fourth several times Ln of X over four -X rays to the fourth power over 16 Plus. We have 81/16. And you can check this using a graph and here we can see the graph of our solution and the slope field of our differential equation. And At the .15 our slope dy over dx is equal to X cube Ln of X. And if you plug in one for X, this will give us one race to the third power times. Ellen of one. That's equal to zero, which is shown here because at this point we have horizontal tangent line with a slope That is equal to zero.

Okay. So to get the original equation um from this differential equation we're gonna want to do is we're gonna want to separate the Dy and the D. T. On the left hand side on the right hand side respectively and then integrate both sides. Okay so to start off let's have DY equals 25 D. T. And the way I did that is I multiply DT on both sides. And now I'm going to integrate both sides to get rid of this D. Y. And D. T. It's on the left hand side. I'm going to have Y. Equals 25 T. Plus. See because that's what you get one. Great and now that's going to be your final answer.

Okay, so we have wise. We could see each five and then distributive Abi, you could be 51 So we have that. Why Prime is equipped to do you have, uh why it should be e five life. But we could wipe time. He could do five. What therefore given in function satisfies given because we can divide life. I am both sides and we get wide with e.


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