I need to solve, in $C[0,1]$, the equation $\displaystyle x(t) - \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds = \sin t$.
Adding the integral part to both sides, I obtain $x(t) = \sin t + \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds$, which is, I believe, a Fredholm Integral Equation of the second kind (yay, Wikipedia!).
However, other than briefly mentioning in class what they are, our professor never really went over how to solve them.
I feel kind of bad asking for a crash course in how to solve Fredholm Integral Equations of the Second Kind (sounds like a bad scifi movie from the early 80s) in $C[0,1]$, but that's essentially what I'm doing...in the context of this particular integral equation of course.
Thank you for your time & patience! :)
Since $$\int_0^\pi (\sin t \cos s) \sin s \, ds = \sin t \int_0^\pi \sin s \cos s \, ds = 0$$ $x(t) = \sin t$ is a solution. Are you worried about uniqueness?
OK, suppose you want the general solution to $$x(t) - \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds = \sin t.$$ Rearrange this as $$x(t) = \sin t \left( 1 + \lambda \int_0^\pi \cos s x(s) \, ds\right)$$ to get that $x(t) = A \sin t$, where $A$ is the number in the parentheses. Plug this back into the original equation to find $A =1$.