I need to find a general solution,

$\displaystyle f$ ∈ $\displaystyle E [0, \infty)$

for the Fredholm intergral equation:

$\displaystyle f(x) = \gamma cos (x) + \delta sin (x) + \lambda (Mf)(x) $

where $\displaystyle M$ is the intergral operator defined by

$\displaystyle (Mf)(x):= \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt$

Where $\displaystyle a$ and $\displaystyle b$ are postive constants and $\displaystyle \gamma, \delta$ and $\displaystyle \lambda $∈ ℝ constants.

I think, that I have to rewrite it:

$\displaystyle f(x)= \gamma cos (x) + \delta sin (x) + \lambda \int_0^{\infty} x^{a-1} e^{-bt} f(t)dt$

And then assuming that λ is a fixed unknown scalar, and so can be incorporated into the equation as follows:

$\displaystyle f(x)= \gamma cos (x) + \delta sin (x) +\int_0^{\infty} x^{a-1} e^{-bt} f(t)dt$

I'm not sure if this is right, and I don't really understand what to do next, can someone help please?