# Heat equation

1. Mar 30, 2010

### squenshl

Consider a heat equation for the temperature u of a rod of length 1:
ut = uxx, 0 < x < 1, t > 0 with boundary conditions ux(0,t) = 0 & u(1,t) = 0. I derived Xn(x) = cos((n+1/2)$$\pi$$x) using separation of variables.
How do I show that $$\int_{0}^1$$ Xn(x)Xm(x) dx = 1/2 if m = n and 0 if m $$\neq$$ n.
I used the product to sum formula: cos(A)cos(B) = cos(A+B)/2 + cos(A-B)/2 to get 1/2cos((n+m+1)$$\pi$$x) 1/2cos((n-m)$$\pi$$x) but I am stuck after that. Someone help, am I even on the right track.

2. Apr 1, 2010

### LCKurtz

Yes, you are on the right track. When m = n use the formula

$$\cos^2(\theta) = \frac {1 + \cos(2\theta)}{2}$$

which is easy to integrate. I'm not sure why you are stuck on the others.

$$\frac 1 2 \cos((n+m+1)\pi x)$$

is just as easy to integrate is $\cos(kx)$.

n and m are integers, you know.