Question

Method of undetermined coefficients...

Answer 1

Must be in the form\[ay''+by'+cy=Ct ^{m}e ^{rt}\]What does C stand for?
I thought it was a constant but the equation\[y''+2y'-y=t ^{-1}e ^{t}\]Does not satisfy this method. i assumed C=1 so it would, bu tit doesn't. Explanation?

Answer 2

@UnkleRhaukus

Answer 3

Find a particular solution to the differential equation\[4y''+11y'-3y=-2te ^{-3t}\]

Answer 4

I believe that the method only applies for \(m\geq0\), but I cannot recall to save my life.

Answer 5

\[4r ^{2}+11r-3=0\rightarrow (4r-1)(r+3)\rightarrow r= \frac{ 1 }{ 4 },-3\]

Answer 6

That makes sense I'm going to look that up. Thanks

Answer 7

@across I found it you are right

Answer 8

since -3 does appear as a root s=1 which is the multiplicity of the root -3

Answer 9

Ok @UnkleRhaukus I could use a little help on my last problem of the night. I posted the question in this stream 3rd from the top.

Answer 10

\[Y _{p}= t(Ae ^{-3t})\]Do I have to include the other root as well?

Answer 11

@Directrix @Hero Can I have a little push in the right direction

Answer 12

@Hero do you see a mistake I have made or is anything coming to you about what to do next?

Answer 13

Sorry, I'm busy doing my own work at the moment.

Answer 14

The answer in the back of the book is \[(\frac{ t }{ 13 }-\frac{ 8 }{ 169 })te ^{-3t}\]

Answer 15

Oh, I understand

Answer 16

@satellite73 @.Sam.

Answer 17

@UnkleRhaukus Just in time I was about to give up. Do you mind helping me with this last question?

Answer 18

where are you up to

Answer 19

I feel like I need to plug in \[Y _{p},Y'_{p}, Y''_{p}\]into the original to solve for A.
I am up to writing an equation for Yp

Answer 20

I feel like I have my roots and s correct

Answer 21

\[4y''+11y'-3y=-2te ^{-3t}\]
\[4r^2+11r-3=0\]\[(4r-1)(r+3)=0\]\[r=1/4,-3\]
\[y_c=ae^{t/4}+be^{-3t}\]

Answer 22

Oh ya I forgot, when you have two diff roots you need to use two different unknowns. What about the t^s? My teacher threatened us if we forgot that on the test

Answer 23

As i understand it t^s, where s is the multiplicity of your roots needs to be out in front.

Answer 24

\[y_p=t\left(c\cdot e^{-3t}+d\cdot t\cdot e^{-3t}\right)\]

Answer 25

Why do you have two terms in the ( ) if m=1 shouldn't there be 1. And I don't mean to come across like I'm questioning you, just trying to undersstand

Answer 26

its a second order equation there will be two solutions

Answer 27

So I see your Yp matches the books except I need to now solve for c and d by plugging Yp into the original equation right?.........Ok

Answer 28

take the first and second derivative of y_p

Answer 29

Right then plug it in. I can do that on my own later. Thanks for your help again. I'm calling it a night

Answer 30

see you next time