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Consicer the initial value problem for the function y; 3yy(1) = 2,t >1.y(t) Transform the differential equation above for y into separable equation for v(t) You ...

Question

Consicer the initial value problem for the function y; 3yy(1) = 2,t >1.y(t) Transform the differential equation above for y into separable equation for v(t) You should get an equation U = f(t,v):Note: In your answer type for v(t); and for t(b) Find an implicit expression of all solutions y of the differential equation above in the form @(t,v) = C where collects all constant terms So; do not include any € in your answer )%(t,v)Note: In your answer type for u(t) , and for -Find the explicit e

Consicer the initial value problem for the function y; 3y y(1) = 2, t >1. y(t) Transform the differential equation above for y into separable equation for v(t) You should get an equation U = f(t,v): Note: In your answer type for v(t); and for t (b) Find an implicit expression of all solutions y of the differential equation above in the form @(t,v) = C where collects all constant terms So; do not include any € in your answer ) %(t,v) Note: In your answer type for u(t) , and for - Find the explicit expression of the solution y of the initial value problem above y(t)



Answers

Solve the differential equation or initial-value problem using the method of undetermined coefficients.

$ y'' + 2y' - 8y = 1 - 2x^2 $

Question three starts with nine. Why? Double prime less. Why? Hee won's e The two X is of course becomes nine are square less. One he holds zero solving For our I get nine r squared equals negative one r squared equals negative one taking the square root of both sides like our square Are are the bulls. Bye. This very negative one night All equals I swear it. If one night is one writing get our equals one there, right which gives me eat the solution, the compliment a solution in the form of Well, that should be an ex C one co sign my one there. Right? My solution on their ex? Yes, C two sign? Yes. And then I have my way. Why? But you won't a e to the two x My pride Why crime equals chain rule. Eat the ex Either the two exes still e to the two X then the derivative. A few exes, too a e to the two x. And why double crime the wolves? Who are you? A. You need to to lex you to extend the same that I think you're a bit of a few X, which is two times two gives me the four. Now, wherever I see the why double prime I'm gonna put for a heated to x times were a I mean with two. Yes, a e for the two x Substituting my purpose My wife eat wolves e to the two x pledges to this equation and substituted. So now I have 36 a e to the X less one, eh e to the two x equals e to the two X They're 4 37 a e to the two x equals Eat to the two X that means thirties Devin A equals one in a equals one over 37. I go back up to my general equation and loves won over 37 you two x.

Here we have a non homogeneous differential equation and we need to solve this by undetermined coefficient. Panther. So first of all right, the auxiliary question for the homogeneous system. So here are square minus two or minus three is equal to zero. And here, middle Thomas negative towards. So we can right here. Negative three r plus r Now factor out are for the first two times and here we can factor out one for the last two times and then we can factor out ar minus three. Therefore, our roots our art is equal to three and r is equal to minus one. So for two, really roots, our complemented solution is given by y. C is equal to see when he was to three ex plus c two years to minus X right now on the right side of the d e or function is cause off two X. Okay, right. So let's assume that our particular solution is in the form off a course off two x plus. Be sign off two X Now take a terry video. So here Why dish is it called toe negative to a sign off two x plus Toby, cause off two X Now take a double dairy with you. So here, avoidable. This is a call to negative for a cause off. Two weeks minus to be sign off two X right now, substitute all these three values in our differential equation. So we're differently. Question is viable. Days minus two. Why this minus three wide is it called toe? Cause off two X But you're wide herbal dishes, Negative for a cause off two X minus four We sign off two X and here negative to multiply by whiteness. But your whiteness is negative to a sign off. Two x last to be cause off two X And here we have a negative three wise, so substitute the value of why. Okay, now simplify this equation. So here we can write. Negative. Say what a minus for me. That is, to your coefficient. Off cause to eggs now. Right, The court coefficient off sign signed to excess. Your four a minus seven B. Now compare the coefficient off course two x And now compare the coefficient off. Signed two x. Okay. So on the right side, we have no sign down. So we have to right here. Zero and by solving both this equation we can get is equal to negative seven by 65 B is equal to negative four by 65. Now put this both constant in our particular solution. So our particular solution is why is it called toe negative? Seven divided by 65. Cause off two X minus four. Divided by 65. Sign off two X And our general solution is given by why C plus y p And now, right, The compliments solution over here. And then right. The particular solution. Right. Thank you.

Question too. I am one devil Prime mine Green. Why ride you will sign of two X. My wider crime becomes R squared minus three are equal. Zero is my equation. Factoring out the r, they are minus three equals zero r equals zero or three. There were my general equation. To start with, C one need to the three x plus c two he to the zero x zero. Anything to the zero powers one. So I have seen one either the three x plus c two. I'm gonna focus my attention on my subpoena equation, and I have wise p of X is going to equal a two x left. The sign to X There were white crime would equal to a negative to a sign to excite captain for that chain rule for trick. So I have to you you cry and then I have That's to be, oh, sign to X and then my wife that will prime derivative of Sinus courtside Negative for a Jane Roe again, who's trying to x minus derivative of coast and negative sign for the sign. U x So in my general, in my wide double crime every time they see double pregnant. Gonna put this equation my double prime. Why? Double prime. It's gonna start off with negative four and sighing u X minus four Sign two X minus three. Why crime is this equation more negative too, eh? Sign? Well, I have two picks. Be thanks equal. Oh, that's distributing my negative three. I have a negative for a why, it's for the sign two x six A sign X line six necks equal flying I'm gonna pull out my co sign and my signs like the ex is the negative for a 96 My sign of two eggs And that gives me negative for B That's six. A home to ex their whole or negative or B plus. It's a has won and negative for a minus six B c on cedar. So negative four B minus six A equals one and thanks. Um B a near 40 Anything for you and negative worry minus its He looks here. So now I saw my system of linear equation, and I am going to multiply the top number by negative three. So I get while b minus 18 a equals negative three. And I'm gonna multiply the bottom. Why two? So get negative. 12 B minus a a equal. See rail. My bees cancel as I get negative. 26 A equals negative. Three A equals three over 26. And then I put that back into my equation, and I have negative forward less six times Read. Actually, I'm gonna use the other equation. Really cool. See road look easier. Negative. Six B minus who were really over 26 row. So negative six bean equals well over 26 and then multiply both sides by a negative 16 And I get being eaten. Make it is Well, thank you to Negative too. Over 26 negative. One teen. So our general equation that we started with age have one e to the three. X c two. Yes. Three 26. Co sign two x less minus one. They're mean. Sign two months. Yes, General

Question or our mission is 12 let's do lie. No. So I have are spared too. Plus two, he see grand actor that do formula with four. That too over a two which it's moving chain plus your which kids? Two. After that, he lives too. I too. So that gives me the form, huh? Why? He would see one, uh, e to the X. Because I haven't I my the ex Who's I nix plus e to the X C to sign And then we continue our work to find out We'll be adding that So I have, um, my general why it's going to be a X Hey X love b see e to the X by wide pearling. Well, then be just a with us c e to the X and my double crime is just Now what I plug that into my formula here have been started on a new change. I get it e to the X did my double problem minus two a class C to the c E to the X less too a X, but be thus c e to the eggs won't necks. Well, I need to the ex so I have seen eat to the ex minus two a minus two Seeing e to the X less to a left To be less to see equals X and then undo it. My aides first. So my aides I'm saying that my ex is first I have 282 a x equals X therefore, to a has to equal one and a equals 1/2. No, I'm gonna get my, uh these So I have my no exes. So I have negative Negative too, eh? Plus two b equals zero My A a negative 1/2 negative. Two times one hat plus two b one be. It was one. And then I have might see e see being so I, um see, I need to the ex equals e to the X Therefore see has to eat one. So now I have all my co affections and I can write out the general of my equation. Though I have my see, I even the experts I could put this in parentheses, but I you to the ex c one co sign X plus Evening X. See you signed X. Yes, I wouldn't have. Thanks. That's one head. Yes, to the ex. Because that was one That is my


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