Pirbavafa, A generalization of Geraghty's theorem in partially ordered metric spaces and applications to

ordinary differential equations, Fixed Point Theory Appl.

1, we have the following correspondence between the eigenvalue problem and existence problem of holomorphic solutions of Heun's

ordinary differential equation (see e.

Elzaki, On the Elzaki Transform and

Ordinary Differential Equation With Variable Coefficients, Advances in Theoretical and Applied Mathematics, ISSN 0973-4554 Volume 6, Number 1(2011), pp.

The second order linear

ordinary differential equation has the general form

In particular if the equation has two independent variables then it can be transformed in an

ordinary differential equation, moreover if an

ordinary differential equation is also invariant under an one-parameter group, then its order can be reduced in one, in the case of first order ordinary equations, the symmetry group can be used to find a solution in terms of cuadratures by two very well known methods, canonical coordinates and the Lie integrating factor.

Determination of the stability of a non- linear

ordinary differential equation by least square approximation.

For

ordinary differential equations and functional differential equations, the existence of almost periodic solutions of almost periodic systems has been studied by many authors.

Numerical Methods for

Ordinary Differential Equation, Wiley, New York.

derivation of new non-standard finite difference schemes for non autonomous

ordinary differential equation2011) that with the physical parameters the system of partial differential equations could be replaced by a system of

ordinary differential equations because both systems gave identical results.

These conditions permit to reduce the nonlinear coupled two-dimensional problem to the two-point boundary value problem for the second-order nonlinear

ordinary differential equation, and then, to a nonlinear equation, with respect to sludge concentration.

It is known that, whereas a first order

ordinary differential equation represents a 1-dimensional problem, a delay differential equation of the form (1.