What Are the Conditions for Solving f(x,t) in the Problem?

[matteo86bo]

Hi,
it is possible to solve the following problem, I mean give at least some conditions on f(x,t)?

$$\int_0^{\infty}e^{-x}[f(x,t)+g(t)] xdx=g(t)$$

 

[matteo86bo]

i give my solution, tell me if there's something wrong with it:

i integrate by parts the rhs:

$$\int_0^{\infty} e^{-x}x[f(x,t)+g(t)]dx= \int_0^{\infty} e^{-x}x f(x,t)dx+g(t) = \newline f(x,t)_0^{\infty} -f(x,t)+g(t)$$

since $$f(x,t)_0^{\infty}=q(t)$$, and rhs=g(t), it follows that

$$f(x,t)=q(t)$$

and then q(t)=0