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Show that the function

$ f(x) = \sum_{n = 0}^{\infty} \frac {(-1)^n x^{2n}}{(2n)!} $

is a solution of the differential equation

$ f''(x) + f(x) = 0 $

$f(x)+f^{\prime \prime}(x)=0$

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so show that function is a solution. That the differential equation after what dash first after equals zero. Okay, so we're first going to find what it is the ruler of FX. So this could be in front zero to infinity would differentiate. This part is not You want off in extra power if two and minus one and two months. One tutorial. So this the first you're over there, Okay. And of course, is from zero. It's not from serious. From what? Because the first time within this zero it becomes a constant so and stuff on one. And secondly, relieve a bit is the first duty of of the derivative of X and they just going to be so when in this one. And it's not a constant, though. So Okay, so it's really stewed from one and next You'LL want to help in extra power to minus two and two minutes to Victoria. Yeah, and heel into the treek. If we change so an x one and chemicals and minus one piece because to zero if we change at Michel's tube zero community, so much want the power. So in the coast, plus one, this is a plus one and extra power to an over two and a pictorial. No. So this is just nothing won Tons and from zero to infinity, eh? That was turf. I'm and extra cult Teo and over two in Factorial. So is this just nothing but that? Okay, so we have shown net a double dash. Now, Which of us a devilish first ethical siro. So that's the solution of the equations.

University of Illinois at Urbana-Champaign