00:01
Okay, we want to solve this equation with these boundary conditions using the separation of variables.
00:07
So we'll call u of x and t is equal to sum f of x times sum g of t.
00:18
If we substitute that in, on the left hand side we'll get f times g dot of t, and that will equal to 4 times g times f double prime of x.
00:34
I'm using primes to mean x derivatives and dots to mean t derivatives.
00:41
I can divide by f times g, so i get g dot over g is 4 times f double prime over f.
00:54
So everything on the left depends on t only, everything on the right depends only on x.
01:02
So that's equal to some constant k.
01:07
I'm going to make it actually minus k.
01:20
So i get g dot is minus k times g, and i get 4 f double prime equals minus k times f.
01:43
Actually i should make one little change on this.
01:49
I'm going to put the 4 onto the g.
01:53
So i get, pause.
02:03
So the reason i'm doing this is i want to put the 4 into the g equation and not into the f equation.
02:27
So this tells me that g is an exponential of minus 4 k t.
02:45
Okay? times some constant out in front, but never mind that for now.
02:50
And then the f equation is the one that's really interesting.
02:57
So we know that f of 0 has to be 0, because for any, if i put in x equals 0 i'm always going to get 0.
03:06
I also know that f of pi has to be 0.
03:11
Okay? so the first one tells me that f is sine of square root of k times x.
03:30
Okay? because it has to satisfy this differential equation.
03:35
We take two derivatives of sine, we get minus the sine back with a factor of k...