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