A Construction of analog of Fredgolm theorems for one class of first order model integro-differential equation with logarithmic singularity in the kernel

The integral representations of the solution manifold for one class of the first order model integro-differential equation with logarithmic singularity in the kernel are constructed using arbitrary constants. The cases when the given integro-differential equation has unique solution are found. The analogue of Fredholm theorem is built for given integro-differential equation. The method of solving this problem can be used for the solving of higher order model and non-model integro-differential equations with singular coefficients.