This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 3 based on Functional equation . Let's give it a try !!
Let be a continuous function with
such that
for all
Find
.
Continuity & Differentiability
Differential equation
Cauchy's functional equation
We are g is continuous function such that for all
and g(1)=1.
Now putting x=y=0 , we get or ,
.
If g(0)=0 , then g(x)=0 for all x but we are given that g(1)=1 . Hence contradiction .
So, .
Now , we can write
(by definition)
Therefore , , for some constant k ,say.
Now we will solve the differential equation , let y=g(x) then we have from above
. Integrating both sides we get ,
c is integrating constant . So , we get
Solve the equation g(0)=1/5 and g(1)=1 to get the values of K and c . Finally we will get , .
But there is a little mistake in this solution .
What's the mistake ?
Ans- Here we assume that g is differentiable at x=0 , which may not be true .
Correct Solution comes here!
We are given that for all
Now taking log both sides we get ,
, where
It's a cauchy function as is also continuous . Hence ,
, c is a constant
.
Now .
Therefore ,
Let be a non-constant , 3 times differentiable function . If
for all integer n then find
.
This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 3 based on Functional equation . Let's give it a try !!
Let be a continuous function with
such that
for all
Find
.
Continuity & Differentiability
Differential equation
Cauchy's functional equation
We are g is continuous function such that for all
and g(1)=1.
Now putting x=y=0 , we get or ,
.
If g(0)=0 , then g(x)=0 for all x but we are given that g(1)=1 . Hence contradiction .
So, .
Now , we can write
(by definition)
Therefore , , for some constant k ,say.
Now we will solve the differential equation , let y=g(x) then we have from above
. Integrating both sides we get ,
c is integrating constant . So , we get
Solve the equation g(0)=1/5 and g(1)=1 to get the values of K and c . Finally we will get , .
But there is a little mistake in this solution .
What's the mistake ?
Ans- Here we assume that g is differentiable at x=0 , which may not be true .
Correct Solution comes here!
We are given that for all
Now taking log both sides we get ,
, where
It's a cauchy function as is also continuous . Hence ,
, c is a constant
.
Now .
Therefore ,
Let be a non-constant , 3 times differentiable function . If
for all integer n then find
.
This solution is incomplete as it assumes that the function is differentiable at zero. Upon taking logarithm of both sides, the functional equation can be converted to a form of Cauchy's functional equation (https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation) and its solution can be inferred from it.