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Abstract The main objective of this thesis is to make a linkage between graph theory and ordinary differential equations with first order, first degree such as (Linear and Bernoulli’s) using simple techniques. Firstly, we transform an orthogonal double covers graphs (briefly, ODCs) to metric graph which we can represent it as matrices. Then, we use these matrices to study the relation between differential equations and the graph theory subject. We use the generalized Fibonacci polynomials (GFP) and the generalized LUCAS polynomials (GLP) to transform the differential equations into a system of equations with undetermined constants. As a conclusion, we solve some numerical examples and evaluate the errors which prove the efficiency and accuracy of the method, we use. |