Question

Find the general solution of y"'-y"-y'+y=2e^(-t)+3. (I got r=-1, 1 after factoring. But what to do next?)

Answer 1

y'" that means you have 3 roots, yours just 2, what is double?

Answer 2

doesn't want to work with me?

Answer 3

1 is double. Sorry, I do want to work with you. I just didn't see you were answering my question.

Answer 4

so, for homogeneous solutions, what do you get?

Answer 5

That's the part I got stucked. It should be c1*e^t+c2*e^-t, but that's not the answer.

Answer 6

sure, that is not the answer since you have a double root.

Answer 7

so far, write out all your roots, no matter what is double or not. they are \(r_1=r_2=1, r_3=-1\), right? you have 3 roots, the homogenous part MUST have 3 solutions. they are\[C_1e^t+C_2te^t+C_3e^{-t}\] because it 's linear, the repeated root must be timed with a variable.

Answer 8

got it?

Answer 9

Yes. But what's next?

Answer 10

ok, non-homogeneous part, you have 2 terms : 3 and \(2e^{-t}\). The partial solution MUST be the sum of 2.
I calculate the term 3 first. let call it is \(Y_p1\)= A

Answer 11

take the first derivative, second derivative, third derivative. add them together to get A =?? Know how to do it? this part is quite easy because it is a constant.

Answer 12

But take the derivative of what? And do you must take up to the third derivative for each problem like this?

Answer 13

you just have \(Y_p1\) =A
you dont have \(Y_p1'\) or higher because they are =0
therefore, the solution for this part is \(Y_p1=3\)that's it

Answer 14

Wait a minute. I must go to the bathroom.

Answer 15

the second part \(2e^{-t}\)
let call it \(Y_{p2}=Bte^{-t}\), why we have to time it with t? because it overlaps with homogeneous part which is \(C_3e^{-t}\)

Answer 16

hahaha... go wherever you want, don't mess up the room.

Answer 17

But the answer has (1/2)te^(-t)+3 after the homogeneous solution. How do I get that?

Answer 18

I don't finish yet. you have \(Y_{p2}= Bte^{-t}\)
please, calculate Y', Y", Y"' . tell me what you get?

Answer 19

But what is Bte^-t? I'm confused.

Answer 20

I shouldn't do this, but it's not easy to explain, save time for both us.

Answer 21

Thank you so much.

Answer 22

yw

Answer 23

next time, feel free to NOT give me medal, I don't want my smartscore up