## anonymous 3 years ago Method of undetermined coefficients...

1. anonymous

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?

2. anonymous

@UnkleRhaukus

3. anonymous

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

4. across

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

5. anonymous

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

6. anonymous

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

7. anonymous

@across I found it you are right

8. anonymous

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

9. anonymous

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.

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. Hero

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

14. anonymous

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

15. anonymous

Oh, I understand

16. anonymous

@satellite73 @.Sam.

17. anonymous

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

18. UnkleRhaukus

where are you up to

19. anonymous

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

20. anonymous

I feel like I have my roots and s correct

21. UnkleRhaukus

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

22. anonymous

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

23. anonymous

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

24. UnkleRhaukus

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

25. anonymous

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

26. UnkleRhaukus

its a second order equation there will be two solutions

27. anonymous

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

28. UnkleRhaukus

take the first and second derivative of y_p

29. anonymous

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

30. UnkleRhaukus

see you next time