# A family of singular ordinary differential equations of third order with an integral boundary condition

## TL;DR

This paper establishes an equivalence between a second-kind Volterra integral equation and a third-order singular ODE with an integral boundary condition, enabling solutions to nonclassical heat equations and analysis of parameter dependence.

## Contribution

It introduces a novel equivalence that facilitates solving certain singular third-order ODEs with integral boundary conditions, advancing methods for nonclassical heat equations.

## Key findings

- Derived explicit solutions for specific nonclassical heat problems
- Analyzed continuous dependence of solutions on parameters
- Established equivalence between integral equations and singular ODEs

## Abstract

We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem.