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(a) Show that the function

$ f(x) = \sum_{n = 0}^{\infty} \frac {x^n}{n!} $

is a solution of the differential equation

$ f'(x) = f(x) $

(b) Show that $ f(x) = e^x. $

A. $C=0$

B. $f(x)=e^{x}$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

So first we're gonna show the function. F packs is a solution to the differential equation. I'm from Essex. Ever bags. All right, so you go first tickets to row there, So there's going to be in from once you infinity on export over in Victoria. Plus, So when n this zero, it is actually one and take Mr if there is going to be and from one to ou infinitely extra power on my sweat or in this one story Oh, and this is just we'll take if they would take care because the man's one So it's gonna be em from zero to in the expert over and over. And Victoria, which is just which is just equal to over afterthe X. So yet we have shown that we have ever that this function is solution to the first equation. I've probably cause f and B show that affect six either power fess. So the exponential function has our solution. All right, so we can just We can just Ah So this differential in clearance, it was the ex and we know that half of zero equals to one wasn't appear. So let's see here. So if we plug in zero on it becomes to one class extra power from one two videos. So it's off off in this zero eso yet We have this initial condition and the differential equation we can find with FX, so we're going to solve it. So this tell us, London Actually, Alex experts some constant and I think awesome. So the final answer is See a constant times a citizens, they want power. Congrats, ethnics. And we just felt in half of zero equals waas. So that implies, Say one equals one. And finally, I have dogs because into power X All right, so everything it's what Rex would have shown that.

University of Illinois at Urbana-Champaign