The problem involves finding real functions \( f \) that satisfy the equation \( f' (x) + f (a - x) = e^x \), where \( a \) is a constant. The context suggests a focus on the Taylor series expansion at the point \( a/2 \) as a potential method for exploring solutions.
The discussion is ongoing, with participants exploring different approaches and questioning the validity of proposed solutions. Guidance has been offered regarding strategies to manipulate the equation, and there is an acknowledgment of the need for further clarification on the role of \( a \) in the solutions.
Participants note the requirement for showing effort in their attempts, and there is a mention of a potential delay in responses due to notification issues. The original poster's use of the Abel Identity is under scrutiny, and there is an emphasis on the need to express solutions without references to \( f(a-x) \).
Please post your attempt, per forum rules.L Navarro H said:Homework Statement:: Find the real functions f so: f' (x) + f (a - x) = e^x
a is a constant
Relevant Equations:: none
.
haruspex said:Please post your attempt, per forum rules.
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?L Navarro H said: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))