Show that this is a general solution of the difference equation

Question

I am currently doing my homework and have been struggling to pass this question:

The difference equation Un = Un-1 + Un+1 is a discrete model for the equilibrium heat distribution along a straight piece of wire running from 0 units to 10 units. The temperature at 0 is 0 and the temperature at 10 is kept constant at 20 C. Show that this is the general solution that satisfies the difference equation for n = 1, 2, 3,..., 9

The answer in the book is: Generalise the sequence that is created by Un = Un-1 + Un+1.

Could someone expand on the question, what is required to solve it?

I think we can use http://en.wikipedia.org/wiki/Recurrence_relation#Solving — lab bhattacharjee, Sep 08 '13 at 14:41
Do you understand the answer that has been provided in the book? — Luke Taylor, Sep 08 '13 at 14:50
it seems to seek the iterative version of $U_n$ from the recurrence one — lab bhattacharjee, Sep 08 '13 at 14:51

score 0 · Answer 1 · answered Sep 08 '13 at 15:07

0

Using this, if we put $U_n=r^n$

we have $r^2-r+1=0\implies r=\frac{1\pm\sqrt3i}2=\cos\frac\pi3\pm i\sin\frac\pi3$

So, $$U_n=A\left(\cos\frac\pi3+i\sin\frac\pi3\right)^n+B\left(\cos\frac\pi3-i\sin\frac\pi3\right)^n$$ (where $A,B$ are arbitrary constants )

$$\implies U_n=A\left(\cos\frac{n\pi}3+i\sin\frac{n\pi}3\right)+B\left(\cos\frac{n\pi}3-i\sin\frac{n\pi}3\right)$$ (using de Moivre's formula)

$$\implies U_n=C\cos \frac{n\pi}3+D\sin\frac{n\pi}3$$

(Putting $A+B=C,i(A-B)=D$)

answered Sep 08 '13 at 15:07

lab bhattacharjee

274,582

...And $U_0=0$ hence $C=$ $____$ while $U_{10}=20$ hence $D=$ $____$. – Did Sep 08 '13 at 15:15
@LukeTaylor, can you complete the answer now? – lab bhattacharjee Sep 08 '13 at 15:19

Show that this is a general solution of the difference equation

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