2. First show that for the completely insulated bar

ut = c2uxx; 0 < x < L; t > 0:
ux(0; t) = 0; t > 0
ux(L; t) = 0; t > 0
u(x; 0) = f(x); 0 < x < L

the separation of variables gives the solution

u(x; t) = A0 + 1 Ancos n x e ( cn )2t
L

n=1 L
X
where
2 Z0 L n x
An = f(x)cos dx; n = 1; 2; 3:::;

L L
then nd the temperature in the bar with L = ; c = 1; and f(x) = x

The height of a birch tree, f(x), in inches, after x months is shown in the graph below. how much does the birch tree grow per month, and what will be the height of the birch tree after 10 months? a. the birch tree grows 1 inch per month. the height of the birch tree will be 16 inches after 10 months. b. the birch tree grows 2 inches per month. the height of the birch tree will be 26 inches after 10 months. c. the birch tree grows 3 inches per month. the height of the birch tree will be 36 inches after 10 months. d. the birch tree grows 2 inches per month. the height of the birch tree will be 24 inches after 10 months.