In the present paper, the analytical solution for an exact Second order fuzzy initial value problem under generalized Hukuhara differentiability is obtained. First the solution of first order linear fuzzy differential equation under generalized Hukuhara differentiability is investigated using integration factor methods in two cases. The Second based on the type of generalized Hukuhara differentiability, the analytical solution of an exact Second order fuzzy initial value problem is derived in two senses of differentiability. Also, several properties for generalized Hukuhara differentiability are obtained on the topics, such as, the univariate fuzzy chain rules, fuzzy integration by parts, among others. At the end, some illustrative examples are presented and these examples show the behavior of the ‎solutions.

In this paper, the existence and the Ulam-Hyers stability of solutions for the implicit Second-order  differential equations are investigated via fractional-orders integral boundary conditions by direct application of the Banach contraction principle.  Finally, we  present some particular cases and two examples to illustrate our results.

The general nonlinear fourth order differential equation, x"""" + f(1) (x,x"x "",x""")x""" + f(2) (x,x",x "")x "" + f(3)(x,x")x" + f(4)(x,x\",x "",x""",t)x = p(t) is considered. It is proved that if the functions f(i) accept certain sufficient conditions, then there exists a bounded space for state variables, in which the equation has periodic answer. For this purpose we use the existence of Green"s function and Schauder"s theorem.