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This book is an attempt to present application of numerical methods in differential equations to undergraduate students, Masters as well as researchers. The chapter describes some basic methods and techniques for programming simulation of differential equations. First I have reviewed some basic concepts of numerical approximations and then Euler's methods. I have provided details on algorithm development using Euler's method. Also I have discussed error approximation and lastly I have used algorithms that an built into the MATLAB programming environment. All the programs could easily be implemented in any programming language such as C, Java. MATLAB is a convenient choice as it was designed for scientific computing and a variety of numerical operations.I have discussed First order systems, Discrete derivative, Euler's method, Evaluating error using Taylor series, programming and implementation by suitable examples. I have also discussed second order systems, Mass-spring, Implementation of Euler's method for second order systems with examples mid-point method. I have also introduced Runge-kutta method. This method is simply a higher order approximation to the mid-point methods.