5

5.#toz1+HHzoHo4o...

Question

5.#toz1+HHzoHo4o

5. #t oz1 +H Hzo Ho 4o



Answers

$$t e^{2 t} \cos 5 t$$

In this video, we're gonna go through the answer to question number five Justin I four. So we asked to right this second order differential equation in normal form on then express that system of different situations in metaphor. Okay, so, first off, let's define a new predator X as first you would say, But why? That means we can rewrite this difference equation as X dash minus reacts minus 10 wide equals sci fi. I think so. The system of equations with this is our first question at this second equation is almost normal. Form is a normal form. We rewrite the second equation. Or rather, the question backs dash as X dash. It was a three x plus 10. Why class society gets in our way. We need to ride this system. Equations in metric forms were looking for on equation like X y dash is equal to some matrix times The vector x y plus Cem vector the correspondent and have a genius bar. So first equation on the right Inside ah of this equation we have three loss of X. We have 10 lots of why on dhe The energy is part is scientist E next up looking at this equation. We just have one loss off X on the right hand side, we turn around. We have zero loss of why on we don't have any in Virginia, it's that's

Our goal for this following problem is to find f prime of A. So we're going to consider FFT, which is equal to T to the 4th minus five kg. Yeah, so when we take the derivative at prime of T, we end up getting that, this is for t cubed minus five. And now we're trying to find F prime of a. Okay, so in finding F prime of A, it's the same thing as if it was prime of Z F prime of two, we just plug in the value of A. For the original value of T. So now we're going to have a four a. cute minus five. I mean, again, if we plugged in, say one, we would end up getting negative one. So that tells us that at the value of X equals one, the slope of this original graph F is negative one.

For the following problem. Our goal is to find a crime of A. So we're gonna start off with a prime of T. Which is Key to the 4th. My next five key. So with this in mind we take the derivative. So we're first gonna find an F. Prime of T. Literacy is before he cubed and it's five. And then we want to find F. Prime of A. So in the same way we would plug in any value do we have here by looking at the movie where he's going to plug in A. T. To end up paying for a Cube -5? Um And this tells us that if for example we plugged in one for a, we would end up getting a negative one. So that means that when A equals one or when X equals one or when T equals one, um Our slope of our original graph is going to be negative one at that point.

We were looking at the lap last transform off t sign to t most applied by the sign off five teeth. Now we know that multiplying by t in the spatial domain corresponds to differentiating and negating in the blacklist transformed to me. So it's a fight. Is this first to calculate the last blast? Transform of signed to tee times five times five t differentiating and negating. This will give us the original back last transport. Now, to calculate this lap last transform, we use a simple, triggered, a metric add entity. You can write the inside here as the coast of three. T minus the coast off 70 all divided by two. And we know that we know what the lack class transform off. A trigger metric co sign function is so we can rewrite this as, um, T is, uh, two. Let's write this size one of the two, uh, t over T squared plus 49 minus t over T Square plus nine. Now, we'll need to differentiate this to obtain the trigger. An electric, the lap last transfer of this function here. Um, that's easily done. Just using the change room. If we apply in the change room when we get this derivative is gonna be equal to first defray chairing the tea terms at the top. Give us this. These terms and then differentiating the denominator gives us the terms at the bottom. Richer. He squared, multiplied by Yeah, won over t squared plus nine squared minus one over T squared plus 49 squares. You know, simple flying. This leads to the expression 20 three s to the power of full plus 58. Most applied by square linus 441. Cool, divided by as to the fourth plus 58 square plus 441 squared.


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