Solve by using power series: y' - y = e^x. Find the...

Solve by using power series: $y' - y = e^x$. Find the recurrence relation and compute the first 6 coefficients ($a_0 - a_5$). Use the methods of chapter 3 to solve the differential equation and show your chapter 8 solution is equivalent to your chapter 3 solution.

Transcript

00:01 Okay, we want to try and solve this differential equation, y prime equals y, with the initial condition y out of zero equals one, using a power series.

00:12 Okay, so what you do is you say you're going to let y equal this power series, which is n equals zero to infinity, a sub in, x to the n.

00:25 Okay, now this differential equation needs a derivative, so we're going to take the derivative of that.

00:31 Really need the p so just can ignore it.

00:34 So we get n equals 1 to infinity.

00:37 Okay, a sub in here it represents the constants or the coefficients of each of the terms.

00:44 So it doesn't change when you take the derivative because it's just a constant.

00:48 And then the derivative of x to the n is in x to the n minus 1.

00:54 Okay, so it says in this differential equation, set the derivative equal to the function so you get n equals 1 to infinity a sub in x n minus 1 equals n equals 0 to infinity a sub in x to the end all right so now what we want to do is we want to manipulate so that we can set these a's equal to each other this is what we're trying to do but to do that we have to get the exponents on the x's to be the same.

01:32 So what i'm going to do over here is i'm going to let m equal n minus 1.

01:40 Okay, so then in place of n, i'm going to put m plus 1.

01:45 Okay, so this one turns into, i need more room, m plus 1 to infinity, a subm plus 1, m plus one whoops m plus one equals one here x to the m equals n equals n equals zero to infinity a sub n x to the n well if m plus one equals one then that means m equals zero okay these variables m and n are just dummies so i'm just going to call this one m also all right now they both start at zero so all we have to do is equate the coefficients the 0th one of this one is equal to the 0th one of this one the first one of this one equals the first one of this one so what we found out is that a m plus one times m plus one equals a m okay or am plus one equals a m over m plus one okay so we don't know what a zero is we'll just say a zero is a but once we know that, then we can find a1...

Question

Solve by using power series: $y' - y = e^x$. Find the recurrence relation and compute the first 6 coefficients ($a_0 - a_5$). Use the methods of chapter 3 to solve the differential equation and show your chapter 8 solution is equivalent to your chapter 3 solution.

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