I have a very simple integral equation I wish to solve, but I cannot put my finger on the appropriate method which isn't overkill for such a simple problem. I feel like I am missing something very simple but I can't figure out what. I have a first-kind integral equation, of the form
$ \int_{-1}^1 (1+at)\, f(t) \, d t = 0$.
I wish to find $f(t)$. I have found literature on general first-kind integral equations but they deal with much more complicated kernels, as well as usually having an inhomogeneity. Is there any analytic, or numerical (quadrature-based) method that works for this type of integral? Or is it a degenerate problem: is $f(t) \equiv 0$ the only solution?
By the way, $a$ is understood to be some complex constant, not a variable. Thanks in advance.
You can rewrite the integral as $$ \int_{-1}^1[P_0(t)+aP_1(t)]f(t)\,dt, \tag{1} $$ where $P_0(t)=1$ and $P_1(t)=t$ are the first two Legendre polynomials. The Legendre polynomials $P_n(t)$ $(n\in\mathbb{N})$ satisfy the orthogonality relation $$ \int_{-1}^1P_m(t)P_n(t)\,dt=0\quad\text{if $m\neq n.$} \tag{2} $$ Therefore, any linear combination $f(t)=\sum_{n\geq 2}a_nP_n(t)$ is a solution to the integral equation $$ \int_{-1}^1[P_0(t)+aP_1(t)]f(t)\,dt=0. \tag{3} $$