Question

Use the method of undetermined coefficients to find the general solution to: y (double prime)+3y (prime)=2t^(4)+e^(t) sin3t.
I got r=0,-3. So the general solution to the homogeneous part is y=c1e^(0t)+c2e^(-3t). I need help finding Y-p

Answer 1

first, can I show you how to type that?

Answer 2

double primes are just quote marks: " single primes are just 1 : '

Answer 3

let me see if I get the same homo answer

Answer 4

\[r^2+3r=0\]
\[r(r+3)=0\]
\[r=0,-3\]
check so now, let's talk about, what is the MOUC

Answer 5

http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx

Answer 6

So, what is the guess for \(t^4\)?

Answer 7

At^4

Answer 8

we remember, it is a polynomial, so we have to include t^4 and all t^n less than t^4, check out that link I attached

Answer 9

Ok. I'm reading it now. Thanks for the hint of how to type y" and y'

Answer 10

It should be At^4+Bt^3+Ct^2+Dt. Is it correct?

Answer 11

yep, ok so now let's deal with the nasty one

Answer 12

e^(t) sin3t.

Answer 13

e^t cos3t+e^t sin3t. I'm not sure about this one.

Answer 14

you forgot constants, make sure you use new letters

Answer 15

now just put em together and what do ya got?

Answer 16

Y (t)=At^4+Bt^3+Ct^2+Dt+K1e^t cos3t+K2e^t sin 3t

Answer 17

\[y_p(t)***\]

Answer 18

but yes

Answer 19

So now I should take the first and second derivative, then plug them back into the equation. Right?

Answer 20

no,

Answer 21

the solution is \[Y=y_c+y_p\] yes?

Answer 22

Don't I need t solve for A, B, C, D?

Answer 23

So the answer should be: Y=C1+C2e^-3t+At^4+Bt^3+Ct^2+Dt+K1e^tcos3t+K2e^t sin 3t right?

Answer 24

ew this is disgusting, sorry I thought it asked for the particular not the total... ok so yes, you will need to take the entire general solution, differentiate it twice and plug it in

Answer 25

That's what I thought

Answer 26

then just like partial fractions you set the bits equal to each other to determine your coefficients

Answer 27

Ok. Great. Thanks

Answer 28

that is nassstyyyy

Answer 29

I recommend variation of parameters to check because that is useless computations

Answer 30

Yep. Gonna do my best. I have another question if you don't mind.

Answer 31

sure just open a new one. I'm working on my analysis hw but I'll look

Answer 32

Ok. Thanks

Answer 33

oh and make sure you tag me