The nth derivitive of f(x) is equal to f(x). What is f(x)?

Question

derivitive with respect to x.

if n=1 and only n=1, then f(x) = e^x

Answer 1

It's more complicated than the previous answers are making it out to be. In general you should have a linear combination of terms like Ce^(wx) where the w is an n-th root of 1, for instance, for n=2 you should get Ce^x+De^(-x), and for n=4 you should get Ce^x+De^(ix)+Fe^(-x)+Ge^(-ix) ... where i=sqrt(-1).

A note to Bruno (underneath my post): You are technically correct about Sin and Cos but they are not needed because if n=4k then i and -i are both 4k-th roots of unity and
(e^(ix)+e^(-ix))/2=Cos(x) and (e^(ix)-e^(-ix))/2=Sin(x)
Now appropriate linear combinations of e^(ix) and e^(-ix) should be able to get you any combination of Sin and Cos.

Answer 2

There are some benchmarks floating round. in case you believe you studied then you definately might want to get sick on your abdomen and also you'll likely merely stick with Intel. then you definately'd ask your self why AMD is popping out with Bulldozer in the first position. you may take convenience in the actual undeniable actuality that the benchmarks are not one hundred% good. i have self assurance those benchmarks come from area of the former u.s. that grow to be the farming us of a and food provide for the former Soviet block. you've gotten to attend with something else human beings... till the end of the week on the most. i'm in my view no longer figuring out to purchase something till techpowerup, guru3d, and anandtech have a evaluation to work out.

Answer 3

TA Timmy has it right. The auxiliary equation for this differential equation is just r^n-1=0. The roots of this are of the form
r=cos(2pi*k/n)+i sin(2pi*k/n) for k=0,...,n-1.
These give solutions f(x) that are linear combinations of
exp(x*cos(2pi*k/n)) sin(x*sin(2pi*k/n)) and
exp(x*cos(2pi*k/n)) cos(x*sin(2pi*k/n))
for all these values of k.

These are, of course, linear combinations of exp(r*x) where the r are given above. I'm just giving the real-valued solutions.

Answer 4

Zero

Answer 5

any linear combination of C*e^x; C real number

for instance f = 3*e^x + 19*e^x
f = 0

Answer 6

f(x) = e^x or f(x) = 0

Answer 7

also, if n=4k, k natural number, then f(x) can be sinx or cosx

timmy, true, i stand corrected

Answer 8

e^x

for n =1

f'(x) = e^x

for n =2

f''(x)= e^x

for n=n

fn(x) = e^x

Answer 9

f(x)=e^x

Answer 10

e^x