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Be Solu Hon Lo oy", hy' Cy XhX lee Y anj be afolution [0 ay", by' Cy X lel Ya then the Sdlution ay" by' Cy -r...

Question

Be Solu Hon Lo oy", hy' Cy XhX lee Y anj be afolution [0 ay", by' Cy X lel Ya then the Sdlution ay" by' Cy -r

be Solu Hon Lo oy", hy' Cy XhX lee Y anj be afolution [0 ay", by' Cy X lel Ya then the Sdlution ay" by' Cy -r



Answers

$\begin{array}{ll}{z^{\prime}+w^{\prime}=z-w ;} & {z(0)=1} \\ {z^{\prime}-w^{\prime}=z-w ;} &{w(0)=0}\end{array}$

We have to determine whether the following is true or false. If the derivative of Y is equal to K times Y and the derivative of Z is equal to K times E, then the derivative of Y plus C is equal to K times Y plus C. So the reason this is relevant to this section, not that anyone cares, is what you're basically saying is that the derivative of a function F is equivalent to some constant times the original function. So the function that satisfies this will actually just be the exponential function. Because if you take the derivative, what happens is the chain rule says that you have to bring this down as a coefficient. So F crime is really going to be K times either the K. T, but either the KT is nothing other than the function, right? That's why it's kind of relevant here. This is just a basic identity. So, let's just take this and and try to manipulate it. So why and Z are both functions of T. I know it looks a little complicated with all this notation, but it basically means that if you plug in a time, this function is going to do something special with it. This function is going to do something special with, it turns out they're going to do kind of something like this. It's not really that special. But regardless, um if you have this condition and you're taking the derivative of a sum of two functions, you can replace it with F and G. I think F and G are in most books, but regardless the derivative of a sum is the sum of the derivatives. So this is what you get. Okay. And then in the if statement, that means that these are both true. Um why prime is K times Y. So this is just K times Y and Z. Prime is just K times E. So you can factor out the cane both of these instances. Can you get exactly what you wanted to show right then? This is equivalent to K times Y plus C. So it just goes to show that this can be thought of as another function called F and F is equal to, or I'm sorry, F prime is equal to K times the original function F. So what I have in brackets right here, that's F. And you're taking the derivative of it, You're basically saying the same thing that I said above networks for the exponential function.

In this video, we're gonna go through the answer to a question of a 17 just 7.5 so as to find the solution in the last space off this ineffable problem. So first, let's take the class transforms both sides. Used the usual theories about transforms off derivatives. Um, that capital, why be the bus transform? Our But what? So why prime what Wydell crime is s where at times, why minus, uh, the initial value of why? Which is just one time by us than minus ah, minus B initial prime value. Which is there? Plus, then we're gonna have s times why minus one minus. Why, a t t o r three? That's in the bill of restaurants. All of that is gonna be three factorial divided by s to the vehicles. One, which is best before. Okay, then we could rearrange this for Why? So that's six over ass to the fore. Then we can add that's a plus one to both sides and divide by the coefficient off one that's s squared close. Yes, minus. Well, let's just rearrange this. So it's gonna be six course s to before, So I guess this one by the top by s to the force we need to force. It was by the bottom. What's the four s squared process? Might as well. And we were only asked this question in the last space so we can leave sleep in there for this question.


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