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solving non-homogeneous difference equations

I'm stuck on these questions and have been trying all day, please help!

a) un=2nun1+2n2+2,u0=1 u_n = 2^{n}u_{n-1} + \sqrt2^{n^2+2}, u_0 = 1

I don't know how to find the characteristic polynomial for this, I get lambda = 1/2^n

b) un=5un14un212n+31,u0=7,u1=9 u_n = 5u_{n-1} - 4u_{n-2} - 12n + 31, u_0 = 7, u_1 = 9

for this, I did the characteristic polynomial: lambda = 1/4 and lambda = 1

G(n)=A(1/4)n+B.1n[br][br]P(n)=nM0+M1n2 G(n) = A(1/4)^n + B.1^n [br][br]P(n) = nM_0 + M_{1}n^2

then when I try and subst. into the original, I get M_1 = 31/3 but M_0 cancels out.

I know I need to find A and B

c) an+2=an+1an+21.2n,a0=10,a1=31+532 a_{n+2} = a_{n+1} - a_n + 21.2^n, a_0 = 10, a_1 = \frac{31+5\sqrt{3}}{2}

I'm assuming this is similar to b) but what would P(n) be?

Thanks
Reply 1
Original post by ~tis
...

Thanks


I can look at this tomorrow evening (bedtime soon) but would you be as kind as to post a photo of the question please
Reply 2
Original post by TeeEm
I can look at this tomorrow evening (bedtime soon) but would you be as kind as to post a photo of the question please


sorry just seen this
Reply 3
Original post by ~tis
sorry just seen this


I need to look up how to do the first one

the second one I just did, bit messy but standard
the "complimentary function" is A + B x 4n (no quarter)

The third one I might try it tonight if I get a minute otherwise on Monday
(edited 8 years ago)
Reply 4
Original post by ~tis
sorry just seen this


I just did the third one (also ok).
Reply 5
Original post by TeeEm
I just did the third one (also ok).


thanks going to try now.

yeah, the first one's got me really stuck because of it being raised to n
Reply 6
Original post by ~tis
thanks going to try now.

yeah, the first one's got me really stuck because of it being raised to n


I have written neat solutions to the last 2, so if you get stuck yell....

The first one I will need to look up a few things before I attempt it which will now be on Monday

all the best
Reply 7
Original post by TeeEm
I have written neat solutions to the last 2, so if you get stuck yell....The first one I will need to look up a few things before I attempt it which will now be on Mondayall the best



I still can't get the answer for b).

I have G(n) = A + 4Bn and P(n) = nM_0 + M_1n^2

Then do I subst. P(n) into the original equation?
i.e. nM0+M1n25nM05M1(n2+1)+4nM0+4M1(n2+2)=3112n nM_0 + M_{1}n^2 - 5nM_0 - 5M_{1}(n^{2} + 1) + 4nM_0 + 4M_{1}(n^{2}+2) = 31 - 12n

but then I get M_1 = 31/3

and I don't know what to do after this
(edited 8 years ago)
Reply 8
Original post by ~tis
I still can't get the answer for b).

I have G(n) = A + 4Bn and P(n) = nM_0 + M_1n^2

Then do I subst. P(n) into the original equation?
i.e. nM0+M1n25nM05M1(n2+1)+4nM0+4M1(n2+2)=3112n nM_0 + M_{1}n^2 - 5nM_0 - 5M_{1}(n^{2} + 1) + 4nM_0 + 4M_{1}(n^{2}+2) = 31 - 12n

but then I get M_1 = 31/3

and I don't know what to do after this


I just finished teaching
Give me a while to eat and I will post (b)
Reply 9
here is question 2
Reply 10
Original post by TeeEm
here is question 2


thank you :smile:
Reply 11
Original post by ~tis
thank you :smile:


no worries...
Reply 12
here is Q3
I must remember to look at Q1

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