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(15 pts) Use variation 0f parameters t0 find the general solution of the given differential equation ty-(+0)y+y=re" 1>0; Given Y;()=l+4, Y()=e (No need to v...

Question

(15 pts) Use variation 0f parameters t0 find the general solution of the given differential equation ty-(+0)y+y=re" 1>0; Given Y;()=l+4, Y()=e (No need to verify the solutions)

(15 pts) Use variation 0f parameters t0 find the general solution of the given differential equation ty-(+0)y+y=re" 1>0; Given Y;()=l+4, Y()=e (No need to verify the solutions)



Answers

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=15 x y$$

Mhm. Give him uh So we are rewriting our differential equation in the form of art squared -2 are Plus one. And this is equal to My next one squared equal to zero. So then it implies that Our it's equal to one. So White Sea of Eggs is going to be cool to see one. He exponent X plus C. Two. E. Exponent X. Experience E. So I have C. Two eggs E. Exponents X. So that's why one it's equal to E. Exponent X. Then why two will be equal to X. E. Exponent X. So then it implies that so it's in place that you have way one way to prime minus Y. Two Y. One time. This will be equal to E. Exponent X. You have X plus one. The exponent X minutes X. E. Exponents X. E. Experience eggs. And this is equal to E. Explaining to eggs. So then this implies that you one prime is going to be x. e. exponent X times E. Exponents X divided by one plus X. Sue the power soon X squared oh divided by he explanations to eggs. And this will give us mine it's X Divided by one plots. Eggs quake. So there are you one you win will be equal to the NC girl of this eggs divided by one plus X squared the ace. And this is going to be called to my next one divided by two. You have lain of one plus eggs suede. Then you too crime. So this implies that our youtube prime will be equal to he exponent X finds he experienced X divided by one plus x squared oh divided by E. Experience to X. And this will be equal to one divided by one plus X squared. So then it implies that are you too will be the inside girl of one divided by one plus X squared the eggs. And this is equal to son invest of eggs. So hank's a particular solution. Y. P. Will be equal to my next one divided by two E. Exponent explain one plus x squared less. Bless X. E. Exponents eggs so invest of eggs. Then a general solution. A general solution, you know solution for a differential equation. Y. X. Will be equal to. It's the power X. Time. See one let's see two X -1/2 lane one plots X squared plus X eggs. Sign invest of eggs as our final answer.

Hello and welcome to problem three of Chapter three. Section 7 you were asked to use the method of variation of parameters to find a particular solution to the given differential equation. Alright so a different equation in this case is Y double prime plus two? Y. Prime plus Y equals three E. To the minus T. And start this off we're going to find the characters to a solution. And that was his drive from the characteristic equation which is R squared plus two are plus one equals zero here we'll have our +11. He squared equals zero. So R. Is equal to negative one will have double it there. So that means why one equals eat the negative T. And white too. Because T. E. It's the negativity. Okay from these uh we'll drive this capital wipes term which is just what is a what we need to add on to the end because we are not left with the equal zero. We're left with an equals E. To the minus T. Okay. And to find this, there's a simple uh equation for this. Well not simple. It's rather long tedious. It's negative. Why one times integral of Y two. GFT were GFT is just this turn here. Over the wrong skin of wise. You can get to that later. D. T. Plus the integral of Oops, Y. two times the integral of why one GFT through the wrong skin dems duty. Okay now what is the run down in this case? Well let's write Ron skin and blue. And that's the two x 2 determinants of its solutions of why 1? Why two by one prime. Why to prime? And in this case This is equal to one times y to prime. Well that's Um either the -T times the derivative of Y two. Get to that in a second. Um Let's go DDT TE -2. Where to subtract this from? Um why to which is to either -1 attempts the derivative of why one prime does negative E to the negative T. So I'm just gonna add like this E to the minus to you. Okay from here we have to find the derivative of T to the minus T. Um There's a few ways to do this. Um But this is really just product rule. So although um Derivative of this is T. 10 to derivative of E- T. is negative. Yeah. The ***. Yeah, negative E to the negative too. And we're going to add this to even the negativity Times The Derivative Tier, which is one. Okay, and if we simplify this some more, we got blue again. Well E to the -2 times either minus two minus T. Either minus two. Yeah. Plus T. E to the -2. T. Keep simplifying. We'll get E to the -2 T minus T. E to the -2 T. Of course this will cancel that with this there. So we'll just be left with E. to the -2 to. Okay. And we can plug everything back into this big Y. Of T. Expression here. Uh right there. Okay so big Y. Of T. Equals negative Y. One. That's a negative view of the -2 comes into rule of Y. Two times. Gmt so Y. Two T. E. To the minus T. G. F. T. S. Three E. To the minus two Over the run skin is E. to the -2 T. Then we're gonna add this to um why too? Uh T. E. To the minus T. Mhm. Who is the integral Of? Why 1? Either? The- T. Times G. O. T. Which is three E. To the minus two Divided by rounds skin which is e. to the -2 T. D. T. Okay from here we'll just simplify and solve for Y. Of T. Uh Looking at this you can see that we'll have an E. To the minus two T. On top and either the mine institute on the bottom. So that will cancel. Cantel cancel, disease will cancel here. Whoops. We'll have Y. Of T. Equal negative et minus T. The integral of T. D. T. And allowed this to T. E. To the minus T. Times the integral of three D. T. Well I'm just going to Um quickly make this 32. Okay and this integral of tea is just one half T squared. So I'll change that one half T. Squared. Okay uh As you can see these two terms can't be combined so we'll have to leave it as is but we want to simplify one more time. Yeah. Oh yes, they can be combined, apologize for that. I will have um three t squared either the minus t minus t squared E to the minus two over to. Yeah. And that will simplify one last time to now. Five t squared E to the minus T over to. So final solution. This entire difference equation, how do we know Y of T is Y equals C1 times either minus two plus C two T to the -2 plus five T squared E to the minus T over to. And that concludes the problem.

Using the method of variation of parameters We have Y. one. So I want to be equal to e. Exponent minus X. Then why to to be equal to e exponent -2 x. So this implies that why one? Why to prime my next Y. Two Y one prime will be equal to minus E. Exponents three minus three eggs. So then this implies that are you one fine? You want crime. It's equal to mine. It's you have signed E. Experience eggs E Experience -2 X. Divided by minus E. Exponents minus three eggs. And this is equal to E. X. Sign mm eggs. So it implies that. So yes this implies that are you one will be the integral of this? So you have the integral of E. Exponents X. Sign eggs the eggs and this will be equal to minus costs into the power X. Then our Youtube prime. So you to crime. It's also equal to sign E. To the power X. It's the power my next X divided by minus E minus reach. E exponent three X. And this is equal to minus is to the part two. X. Sign E. To the power eggs. So to find you too. Youtube. You too is going to be the inti girl of sign. Each. The power X. You have eaten the part my next ex. So it's signed two will be the integra. Thank you girls minus E. To the power to X. Ein eat the power eggs the eggs. And this is equal to eat the power X. Cause it's the power eggs mine. It's sign it's the power Thanks. So then this implies that particular solution will be equal to minus E. To the far my next ex. Because it's the power X minus E. To the power minus two X. You have sign U. To the power X. Mine. It's is the power X. Cause into the power X. So then this implies that our general solution a general solution to a differential equation Y. X. It's equal to you have see one minus cause it's the power X. Times you to depart my next eggs. Bloods you have C. Two minus sign each. The power X. Plus eat the power eggs. Because because E. To the power X. All times E. To the power minus two eggs as a final answer to the differential equation.

Even you have what I want. I want to be equal to Each the power -2 x. Why to its equal to X. It's the power minus two eggs. So then this implies that's why one Why to crime mine? It's why do I want prime would be equal to? It's the power minus four eggs. Something Y. You want crime? It's equal to minus E. Experience minus two X. You have eggs. He exponent -2 x. Divided by X. to the Power three. He explain it's minus for thanks. And this is equal to my next one divided by X squared. So then it implies that are you one is going to be the insignia of when divided by X. Squared the eggs. And this is equal to X. To the power my next one or 1 divided by X. And you too. You're super prime. It's equal soup. It's the Power -2 x. You have each the par -2 eggs Divided by X. to the Power three. Used to the par -4 eggs And this is equal to one divided by x cubed. So then it implies that you two will be the insider of one divided by sq the eggs. And this is equal to -1, divided by two X. Quaid. So dan a particular solution Y P X. Is going to be equal to eat the power minus two eggs divided by eggs. Mine. It's eggs E to the car minus two X, divided by two eggs squared. And this is equal to It's the Power -2 x. Invited by two X. Then a general solution would then be so I Jenna a general solution Jenna solution Y. X. It's equal to it's the power minus two X. Sometimes you have c. one Plus c. two x plus one Divided by two x. As our final answer.


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