Some Classes of First-Order Integro-Differential Equations and Their Conjugate Equations.

The paper is devoted to the investigation of some classes of first-order integro-differential equation with singular kernel and its conjugate equation. The kernel of non-conjugate equation has singularity at the point , and the kernel of conjugate equation has singularity at the point . First, two conjugate classes of functions are introduced and then in these classes of functions the model equations are investigated. Further in three cases, in depending of the roots of the characteristic equation, the solution of studied equations were found in an explicit form. It is shown that the general solution of investigated equations contains some number of arbitrary constants. Also, there were found the case when the model equations has only unique solutions. Also in this work we studied the non-model equation. When the roots of the corresponding characteristic equation are real and different the solution of the non-model equation were found through the resolvent of corresponding Volterra-type integral equation. [ABSTRACT FROM AUTHOR]