If y(t) = 4e-t4e 5t is a solution to the following equationy"' + py + qy = 0,what are p and q?pq =...

Find particular solutions. $$\frac{d P}{d t}=P+4, \quad P=100$ when $t=0$$

Okay, So what we have here is a differential equation wherein were asked to find its solutions. Given the condition, which is a physical 20 Q is equal to speak. So what we want to do first is to multiply both sides my one over. Q. By doing that, we now have pick you over. Q dp Is it closed with one over fight now we want the market. I would say it's by. BP are assaulting Equation is thank you over. Q. Is it going toe 1/5 b deep? So taking the integral of both sides we have. And then, if you is include the 1/5 be let's see. So we can raise both sides to G and remember that See, here it's just a constant. It can take any value. Therefore, our assaulting equation is Q is equal to C E raise. Don't 1/5 feet. So this is now the general solution off our differential equipped should. So we want to find its particular solution. So we make use of the emission conditions given away to go to so far enough, understand? So a B is equal to zero. Que is able to speak so situating these So our differential, our general solution immune. So 50 is equal to see erase toe 1/5 0 So this will just be equal. The one therefore our see here is a photo 50 writing the whole equation for its particular solution. We have que is equal to 50. It is the 1/5 beat. So this is the particular solution and our final answer.

Mr So we're giving her equation P equals peanut. Mhm. Times E. To the art times t minus T. O minus five. Right? So obviously the first step Would be to move five over to the P. cool. All right, so we get P plus five equals P. Not times E. To the R. Times T. Honesty. Not now. We still have this peanut upfront. Remember we're solving for T. Uh Not even T. Not just T. So we can move this peanut down here with divisions. So therefore we get P plus five over peanut equals right? Since we have this our times t minus T. Not. We can distribute that to make our lives a little easier. We E. To the R. T. Um And then minus R. T. Not but we can rewrite that like this right? Due to exponential property. We can rewrite it like that. Um Okay so now since again we're solving for tea we can take this guy here, right? We're not worried about t. Not so we we can move this over actually by multiplication right? We can multiply on and each of the positive artie not which would cancel this out. And then just move it over there. So we would get E. Positive artie not times P. Plus five over peanut. Oh that equals E. To the R. T. Okay. Um Okay now last but not least we have to get rid of this E. Uh And this are of course. Um So the only way you can really do that is go Ellen right? Because Ellen is the one thing that will cancel any Ln we do that to both sides. So now we get our team moved out to the front. L. N. E. But the L. Any cancels. Right? So we just have our tea times Ln um of E. To the er T. Not times P plus five over peanut. All right, now, last but not least. Right. We have this are up front. So therefore we just to solve for T. We just say one over R times Ln. Of E. To the RT not times P Plus five over Peanut. All right. And that is it. Um We have solved for tea. And so now we are all done.

Okay, So what we have here is a differential equation. Where inner asked if I this particular solution given the initial conditions at physical to serial, few physical to 50. So we are going to look at our B shot. We can further simplify these US The U over DP is a quote the 0.3 I'm skew Mind those 400. So this is now a separable differential equipped should So we can rearrange this as speak you over Q minus 400 is equal 2.3. The heat getting the integral of both sides will get and off que minus 400 is equal to three p. Let's see. So we can further the 65 Thies to get off. I salute you and what we're going to do next is to get the power off the port sites. Now we'll have theories. Ellen off U minus 400 is equal to erase to treaty that see, So we know that the left side of the equation just be equal. Tokyu minus 400 And the sea year is just constant. So you can take any value we can write the right side us see erased three feet for further isolate. Q. We touched for the negative 400 to the right science, and now we have. Q is equal to see based on three D plus 400. So this is now the general solution of our differential degree. Should So what? We're asked us to fight, its particular Sidhu showed. And they do this by computing for the value. Upsy so we make use of the initial conditions given a while ago, which is at B, is it about zero? Hugh is equal to speed. So we substitute these to our general solution will have 50 in sequel to see a raise to three times zero plus 400. So this will just be equal One and they will have seen is equal the negative 3 50 Rewriting the Holik we showed we now have que off. Being is equal toe for hundreds minus 3 50 e Today's the day. So this is now the particular solution

All right. The problem we wish to solve a given differential equation, DQDT equals 2/5. This question is challenging understanding of how to solve differential equations, but we know the initial conditions Q of zero equals 50. Just all such a differential equation, we're going to be using the separation of variables method. The separation of variable method requires three steps and step one will isolate our terms are Q and R. Two terms on either side of the equal side. That is the separation of variables. And step to integrate both sides. And in step three we solve our newfound equation for the initial conditions given. So in step one we write D Q over Q equals D T over five, integrating gives integral over Q equals integral. DT over five. This solves as L n Q equals T over five plus C. Which further solve this, Q equals 80 to the T over five. Yeah. Thus talking international conditions we have 50 equals 82. 0 or 50 equals eight and one thus A. Is simply 50 from this. We obtain our final solution for the initial conditions, Q equals 50 E. To the T over five as is highlighted at the bottom.