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[25 pts] Five differential equations (a-e) are listed below: For each differential equation; choose from the list of functions (A-Z) the appropriate form Of the par...

Question

[25 pts] Five differential equations (a-e) are listed below: For each differential equation; choose from the list of functions (A-Z) the appropriate form Of the particular solution Yp(x) if you were to solve for Yp using the method of undetermined coefficients: dy dx y = 9 cosx#-& =T2xe 4 4y = 7e2r 4 y = 6x dy dx? y = 3x sin(2x)A. AxB. Ax + BC. Ar- + BxD. Ar" + Bx + €E. Aer H Ac4E Are*G. (Ar + BJe*J: (Ax + BJezxAcos x B sin xAx cosx Bx sinx(Ax B) cos x (Cx D) sin xN: Acos(2x) B sin(2x

[25 pts] Five differential equations (a-e) are listed below: For each differential equation; choose from the list of functions (A-Z) the appropriate form Of the particular solution Yp(x) if you were to solve for Yp using the method of undetermined coefficients: dy dx y = 9 cosx #-& =T2xe 4 4y = 7e2r 4 y = 6x dy dx? y = 3x sin(2x) A. Ax B. Ax + B C. Ar- + Bx D. Ar" + Bx + € E. Aer H Ac4 E Are* G. (Ar + BJe* J: (Ax + BJezx Acos x B sin x Ax cosx Bx sinx (Ax B) cos x (Cx D) sin x N: Acos(2x) B sin(2x) Ax cos(2x) Bx sin( 2x) P (Ar B) cos(2x) (Cx + D) sin(2x) NA - undetermined coeflicients is not #pplicable None of the above; the method applies but the form of yp is not in the list Are"x



Answers

Obtain a particular solution of each differential equation. (Use the results obtained for Problems $5-10 .)$ $$ \frac{d y}{d x}=\cos x \frac{d y}{d x}-y \sin x ; \quad y=\frac{1}{2} \text { when } x=\pi $$

Hello the level you are going to solve the problem number 44. For the first order linear differential equation. Ship of Bernales equations. Okay, so we have to get the general solution for the following the french equation. X. Dy mighty X plus two. Y equal. Bye. Sign X over X. Okay, so for number eight the by using case or C. S. So that or any device. So we can just get the general solution for that differential equations. So it will be y equal see at the constant -5 design X over X squared. And that's a general solution. Okay, for number B. Okay. For number B the particular solution that satisfies that boundary condition. Why? Oh boy equal zero is that's why you will be equal negative boy minus phi cuisine X over X squared. Okay, so we saw substitute would see equal. Bye. Okay. Okay, so for number see okay, we have to get the graph of the particular solution over the interval boy over to and five boy over to. So it will be as soon. Okay, so that's like half of the main uh that differential equation of the particular studios. Okay, Thanks for watching. And see you see you in the next differential equation. See you later.

Hello. Everybody who are going to solve Probleble # 43 four. Uh First order linear differential equation. Chapter of Bernoulli's equation. Okay, so we have that govern differential equation dy by the X plus three Y. Over X. Equal sign X. Okay, so we have to get the general solution of that. The french equation by using say S. Or cast. Okay, so the general tradition will be equal Y. Equal oh, negative X. To the power three, cuisine, X plus three X squared sign X minus six. Sign X plus six. X. Cuisine, X plus C. That's all over X. to the Power three. That's For number eight. Okay. and four number or requirement number B. Okay. For number 30. The particular solution that satisfied the boundary condition. Why why Equals 0? So it will be as why equal negative X. To the power three cousin X plus three, three X squared sign X minus six. Sign X plus six X. Cuisine X plus six. Boy- Boy. To the Power three. And that's over. Okay. Over X. To the power three. And that's for number B. And four. Number C. Okay, we have we have to get the graph of the particular solution Over the interval pi over two and 5 pi over two. So it will be shown as below. Okay, that's poor number. See that's in terrible between bye over two and five by over two. Okay, so we can chew as a follow. Okay, thanks for watching. And we thank you for the next differential equation. See you later

Talk about this question. So we have to find a solution of the differential equation. So let's move all the terms with X over to one side. With a separate variable. So we have uh uh costs Y and D. Y. Or were on one side. And let's take the exit toward the other side. So too is a constant. So we have two times T. X. Or X. Now we can easily integrate distance. The variables are separated. So the integration of course is science. So we have signed Y. We have signed Y. And this is equal to two comes out. So we have the X over X. S. Natural log of X. Or absolute value of X plus the constant of particular C. Uh Not to find the value of the constant in particular. We are going to use this particular relations. So at Y equal to pi over three which means that why why is replaced by power three Xs minus one. So absolute value of minus one Plus C. So sign by over three as root tree absolute value of minus one is one and natural log of one is zero. So C come sodas, route three. So uh three root tree comes over here so we get we get, the final answer is signed. Y is equal to two times natural log of X plus rural tree, and this is the final answer. Yeah. Thank you.

So you'll notice that I started solving this problem by rewriting the left side of my equation, As do I. D. X. And I did this because in my next step, I'm going to multiply both sides of my equation by D. X. And this is going to give us a separation of variables that will allow us to integrate each side of our equation. So on the left side of our equation, we have do y, and this is equal to sign X E to the coast on X Power D X. And so from here we can go ahead and integrate both sides of our equation. I'm on the left side. We're just going to get why and on the right side, we're going to have to use U substitution. So for now, I'll rewrite this as the integral of sine x times e to the coastline ex the ex. And, uh, we're going to go ahead and use the inside of this e exponents, as are you. So I'll do that to the right here. If we have you is equal to co sign X, then that tells us that d U is equal to negative sign x, d x And so from here we can go ahead and substitute this into our integral We get why is equal to the integral of negative e to the you power. Do you? And so when we integrate, we're just going to get wise equal to you Negative e to the you power plus c and when we substitute are you back in? We find that our general solution is why is equal to negative e to the coastline X power plus C.


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