Find the function

[George Saliaris]

Let f: R ->R be a differential function such that :

1) f(0) = 1.

2) f'(x) =e^(2x)*(f(-x))

.... What I have done so far :

1) I proved that f'(-x) * f'(x) = f(x) * f(-x)

2) I "suspected that f(x)= e^x", so I am trying to show that f'(x) = f(x) so then we will have f(x) = c*e^x with no success so far. Is there something I am missing?

 

[pka]

What is the derivative of \(f(x)=\dfrac{1}{2}e^{2x}+\dfrac{1}{2}~?\)

 

[George Saliaris]

Rhetorical, but it's e^2x..

 

[Steven G]

Rhetorical, but it's e^2x..

Actually it is e^(2x). So does the function given by pka satisfy your conditions?

 

[George Saliaris]

Eh not really since f'(x) = e^(2x) which is not equal to e^(2x)*f(-x)= e^x

 

[George Saliaris]

I have been writting pages and pages and so far nothing just useless stuff.Can someone give me an insight?

 

[Deleted member 4993]

I have been writing pages and pages and so far nothing just useless stuff.Can someone give me an insight?

Can you make sure that the initial condition is correct?

 

[George Saliaris]

f'(x)= e^(2x) * f(-x) for every x real, f:R->R ,f(0)=1. f(x)=? f is differentiable not differential.