### ODEs

*Ordinary Differential Equations*(ODEs) typically result directly from Newton's laws of motion, restated here as follows:

This is a second-order ODE relating the force on a mass at time to the second time-derivative of its position ,

*i.e.*, . A physical diagram is shown in Fig.1.1. From this ODE we can see that a constant applied force results in a constant acceleration , a linearly increasing velocity , and quadratically increasing position . The initial position and velocity of the mass comprise the

*initial state*of mass, and serve as the

*boundary conditions*for the ODE. The boundary conditions must be known in order to determine the two constants of integration needed when computing for . If the applied force is due to a spring with spring-constant , then we may write the ODE as

(Spring Force + Mass Inertial Force = 0)

This case is diagrammed in Fig.1.2.
If the mass is sliding with *friction*, then a simple ODE model is given by

(Spring + Friction + Inertial Forces = 0)

as depicted in Fig.1.3.
We will use such ODEs to model mass, spring, and dashpot^{2.6}elements in Chapter 7.

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Formulations