5

+ l)ez?+2x 1 Given that y = (c- and y(0) = find y_ 2...

Question

+ l)ez?+2x 1 Given that y = (c- and y(0) = find y_ 2

+ l)ez?+2x 1 Given that y = (c- and y(0) = find y_ 2



Answers

Solve the differential equation.
$ y' + y = 1 $

The topic of this question is linear differential equations. This question asks us to solve this linear differential equation. Mhm. Since we noticed uh it's a first order linear differential equation, since the highest derivative is the first derivative. Yeah, we know that this equation can be solved using the method of integrating factors. What do we mean by that? Well, we mean that our goal is going to be to write the right or the left side of the equation as a product rule. So the derivative of some product uh uh let's say why is a function of X? Just to be clear, the independent variable is X. Okay, anyways, so that I can write D. D. X. Of some product. And what is going to be or what is that product going to be? Well, we almost have a product rule here. It looks like we have. Why times what could be the derivative of some function was the derivative of Y. Plus what could be? Yeah. Where the function goes, Obviously the derivative. This is not a product rule. Since here we would have to have a function whose derivative would be one. And so one way to get a function like that. Ah to get a function here whose derivative would be here is by multiplying every term in this equation by a certain function. No. If we multiply that function, let's call it uh F A barracks, then we would want the derivative of ffx to be itself ffx. So what function do we know? Does that? Well, we know the exponential function is its own derivative. And this exponential function is indeed the function that would allow us to write this as the derivative of the product uh F of X times Y. So our differential equation would become this all we've done is multiplied by F. Of X equals E. To the X. On both sides. And since the derivative of earth in this case is F. Itself, um we can right rewrite the equation. This left side here is F. Y. Prime plus F. Prime Y. Which is this product rule Now we can simple simply integrate both sides with respect to X. Yeah, on the left side we have the integral of the derivative. Or, well, yes, we want to find the anti derivative of the derivative of E. To the X. Y. Which is just E. To the X. Y. Yeah, you did the X times wine. Yeah. And on the other side we have the anti derivative of E. To the X. It's either the X plus the constant of integration to account for the family of anti derivatives that make up this indefinite integral. So here we have our equation. We are almost done. We just want to write it as an explicit function of why? So have why? On one side by itself. So just divide out E to the minus X. So this becomes one, this becomes C, times E to the minus X. There we go. This is our solution. It's a general solution since there is a constant constant of integration. And if we were given initial conditions, such as Why is equal to three when X is equal to one, then we can substitute that into find which exact or which solution precisely satisfies that initial condition. That brings us to the end of this video. Thank you for watching.

Hello. We have to solve the given differential equation that is G. T. Y. Please develop breast away. It cost a hero. So. Exalted form of the seclusion and mr plus M press to cost +20 So the complexity of this will be minus B. Plus money. Scuttled off one monies for into into one upon to. So this will because two miners of one x 2 plus minus. You squirreled off 78 upon 12 ι.. So alpha is one way to beat is square root of seven. So the solution can bitterness into the power of Halifax Stephen cause she went side we tax plus you do cause be text. This is the solution. So this will be cause to the to the power minus X. Y. Do seven off sign Beat. ISAT scores out of seven x 2 of X. Plus. She too calls scurried off seven x 2 or Fax. So this is the and so I hope you understood.

We have y prime plus two y equals one. Now, let's go ahead and figure out the integrating factor. Each of the two t now multiplying through by the integrating factor and then rewriting the left hand side using products role we get into the to T Y Andi to the to T. Okay. Okay. So again, we have multiplied through the differential equation by each of the two t and then we recognize the left hand side is equal to what I have written here. Okay? And now let's find the integral of each side. Yeah, divide through by each of the two t. That's it.

But we have y prime plus two y equals one. Let's go ahead and figure out the integrating factor. So now we're gonna go ahead and multiply the differential equation through by E to the to T. And then we're going to rewrite the left hand side using product rule. So the left hand side is gonna look like derivative of each of the two t y. On the right hand side is each of the two t. Let's go ahead and integrating side. This gives us this. Now we're almost done. But we still need to solve for the value of C. So this is what why is equal to We just need to figure out the value, see, and we know that why had 001 It's also equal to one half plus c. So we concluded that C is equal to one half. That's it


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