Many problems in physics are described by differential equations. This is due in part to the basic laws of nature (like Newton's second and Schrödinger equation) being differential in form,

The differential nature of these physical laws in turn may be a reflection of our use of continuous variables like position and probability. (The use of differential equations may also reflect traditionally-trained physicists viewing problems in differential forms). As a complete discussion of differential equations is beyond the scope of this chapter we will deal only with linear first and second order ordinary differential equations. We start with deriving two methods to solve first order differential equations numerically (Euler and Runge-Kutta). These methods can be extended to solve second order differential equations which we will do using the harmonic oscillator and the realistic pendulum as examples.

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Hans Kowallik, Manuel Paez