00:01
We're asked to solve this differential equation, y prime minus y equals zero, using power series.
00:06
But first of all, we can see that the solution is just an exponential function.
00:10
And so we should, you know, we can get that.
00:13
But to use a power series, we should basically wind up with a power series that is a representation of the exponential function.
00:21
So i use the summation notation here, because we really only care about this lower bound of the summation, the upper bound always goes to infinity.
00:31
And we always are summing over n.
00:33
So i just use this simpler notation to clean that things up.
00:37
So this is the summation of n goes from zero to infinity of c to the n, x to the n.
00:41
So we assume that for y.
00:43
Take the derivative, but we get summation from n equals 1 to infinity of c to the n, and x to the n minus 1.
00:52
So what we have here, and we could change this to 0, but it doesn't really make a difference.
01:00
Because if n was zero then this would be zero and it wouldn't add anything to the summation take these two things i think these two expressions and plug them into here and we get this now what we wanted to do was we want to get everything we want to get x all these xs to the same power so we can factor them out so we can shift n so we can shift n by add 1 to n so we get, you know, basically let n go to n plus 1.
01:32
So we get c to the n plus 1, n plus 1, n plus 1, x to the n, and now we're going from zero to infinity.
01:40
And notice that, you know, that is the same.
01:44
These two summations are the same...