Function defined by a derivative Watch

psanghaLFC
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#1
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#1
The question is

In the region |x|<=a, y(x) is defined by the differential equation dy/dx=f(x), where f(x) is a given even, continuos function Prove that

y(-a)-2y(0)+y(a)=0


would that just follow from

integral(a to 0) of f(x)=integral(0 to -a) of f(x)?
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generalebriety
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#2
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#2
Sounds right to me.
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psanghaLFC
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#3
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#3
and isn't y(0)=0?
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DFranklin
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#4
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#4
Why would it be 0?
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psanghaLFC
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#5
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#5
if it's derivative is an even function doesn't that nesseserily imply y(x) is odd and so

y(x)=-y(-x)

y(0)+y(0)=0?
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DFranklin
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(Original post by psanghaLFC)
if it's derivative is an even function doesn't that nesseserily imply y(x) is odd.
No.

You might want to think about a concrete case. You might as well make it simple, and consider the case where f(x) is just 0.
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psanghaLFC
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yeah i see
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psanghaLFC
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#8
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#8
would it be correct to say "either y(0)=0 or y(x) is a constant function" because under both conditions that result will hold
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DFranklin
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No.
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psanghaLFC
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#10
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#10
So, if y(x) is an even function but not a constant function then y(0) doesn't nesseserily have to be zero? Can you given me an example?
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DFranklin
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You mean dy/dx is an even function. Yes I can give you an example, but I think you should think about it some more first. You are forgetting something absolutely fundamental when it comes to solving differential equations.
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psanghaLFC
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#12
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#12
constant? Ahh yes f(x) can be an even function and still statisfy d/dx y(x)=f(x) and d/dx(y(x)+c)=f(x)
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psanghaLFC
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#13
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#13
The next part says prove that

integral (a to -a) y(x)=2ay(0) so

since y(x) doesn't nesseserily have to be even y(x)-y(0) does.

So

the result follows from

integral (a to 0) (y(x)-y(0))+ integral (0 to -a) (y(x)-y(0))=0 correct?
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DFranklin
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I don't know what you're trying to say:

integral (a to 0) (y(x)-y(0))+ integral (0 to -a) (y(x)-y(0))=0

is essentially equivalent to

integral (a to -a) y(x)=2ay(0)

So you would need to actually prove that integral (a to 0) (y(x)-y(0))+ integral (0 to -a) (y(x)-y(0))=0
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psanghaLFC
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#15
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#15
Yes but y(x)-y(0) is an even function right? So the integral of y(x)-y(0) from a to 0 is the same as from 0 to -a. Is that correct?
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DFranklin
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#16
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No, it isn't an even function. And what ever it is, you need to prove it. (i.e. not just say "it's an even function, right?")
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psanghaLFC
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#17
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Oh, i mean an odd function. Is that correct, even though i need to prove it?
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DFranklin
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#18
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Yes.
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