E-StoreStratergies for solving functional equations.
This articles tries to explore some ways to solve functional equations.It also has some standard examples of functional equations.
FUNCTIONAL EQUATIONS:
In mathematics or its applications, a functional equation is any equation that specifies a function in implicit form [1]. Often, the equation relates the value of a function (or functions) at some point with its values at other points
STANDARD FUNCTIONAL equations with solutions.
f(x+y) = f(x)f(y) Solution = a^tx where a and t are some parameters
f(xy) = f(x) + f(y) solution = t*lnx
f(xy) = f(x).f(y) solution = x^n
f(x+y) = f(x)+f(y) = t.x
f(x) is a polynomial of degree n
f(x).f(1/x) = f(x) + f(1/x) then f(x) = 1+-x^n
Example:
f(xy)=f(x).f(y), f(2) = 8 find f(x)?
ans: The solution to this functional equation is x^n.
using the give value we can guess the function.
f(2) = 8
2^n = 8 => x = 3
therefore the function is x^3.
SOME NOT SO STANDARD FUNCTIONAL EQUATION.
there are of two types.
You might find some other type problems too .. But Given the limit of length of the article I can only explain the important three.IIRC there are some 2 ~ 3 more examples available.
1) f((x+y)/n) = something
2) f(xy) = something
Our main statergy for solving such functions is to obtain f'(x) and integrate it to obtain f(x).
F'(x) can be obtained in two ways.. Either directly derivating the give functional relation(ex: f(xy) = something)
or using the limit concept (lim h->0 (f(x+h)-f(x))/h)
I will explain each of the three with examples.
1) f((x+y)/n) = something
q)f((x+y)/2) = (f(x) + f(y))/2
derivating treating y as constant.
f((x+y)/2)*1/2 = f'(x)+0/2
replacing x by 0 and y by 2x
f'(x) = f(0) = -1
Integrating we have f(X) = -x +c.
putting x = 0 we get c= 1
f(x) = -x+1
2) f(xy) = something
q)f(xy) = f(x)f(y) f'(1) = 1
f'(x) = lim h -> 0 f(x+h)-f(x)/h
f(x+h) = f(x(1+h/x)) using this idea compute the limit.
we get it as f'(1)f(X)/x
integrating f'(x)/f(x) = 1/x.
f(x) = x
There are many more examples of this sort in JPNV calculus by skgoyal.
in the chapter defrientiablity and continuity.
I have done lot of research on this particular topic. I have many more problems That I can show but typing problems like this is very difficult.
I agree that I am still a noob in this topic and I'm trying to catchup.
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