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

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