# Isn't it still 2016?

Calculus Level 3

Consider the functional equation, $$\dfrac { { d }^{ 2017 }f(x) }{ d{ x }^{ 2017 } } =f(x)$$ for a function $$f$$ which is defined for all real.

Let the solution to this equation be of the form of: $$\displaystyle \sum _{ i=1 }^{ 2017 }{ { c }_{ i }{ e }^{ { r }_{ i }x } }$$ where $${ c }_{ i }$$ are arbitrary constants independent of $$x$$.

Now, given that $${ r }_{ 2017 }=1$$ and $$\left| \displaystyle \sum _{ i=1 }^{ 2016 }{ { r }_{ i } } \right| =S$$.

Also, $$\displaystyle \lim _{ x\rightarrow 2017 }{ \frac { { e }^{ 2016S-x }-{ e }^{ -1 } }{ 2017-x } } 2017=A{ e }^{ -a }$$.

Then calculate $$A-a$$.

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