What is the Taylor series expansion at ##a/2## for finding real functions?

[L Navarro H]

Homework Statement

Find the real functions f so: f' (x) + f (a - x) = e^x
a is a constant

Relevant Equations

none

Mod note: Member warned that some effort must be shown.

 

[haruspex]

Homework Statement:: Find the real functions f so: f' (x) + f (a - x) = e^x
a is a constant
Relevant Equations:: none

.

Please post your attempt, per forum rules.

 

[L Navarro H]

Please post your attempt, per forum rules.

I use the Abel Identity to find the functions, but I'm no sure if it's correct to used it in this problem
my answer is {e^x , - 1/(3*e^(2x))

 

[haruspex]

I use the Abel Identity to find the functions, but I'm no sure if it's correct to used it in this problem
my answer is {e^x , - 1/(3*e^(2x))

Are you saying that each of those is a solution to the equation? Doesn’t look that way to me. Shouldn't 'a' figure in the answer?
Please post your working.

Hint: you need a strategy for eliminating references to f(a-x) and its derivatives so that all the references to f are to f(x) and its derivatives.

Btw, sorry for the delay.. for some reason I did not get the alert. Happens sometimes.

Still there @L Navarro H ? If you need a stronger hint, try two things...
1. Write the equation swapping x and a-x.
2. Differentiate the equation.
Can you see how get rid of references to f(a-x)?

 

[wrobel]

have you tried to expand a solution in the Taylor series at the point $a/2$? That is to look for solutions in the form $f(x)=\sum_{k=0}^\infty f_k(x-a/2)^k$

upd: but haruspex's idea is better