2: Second Order ODEs
- Page ID
- 89121
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"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel (1906-1978)
- 2.1: Introduction
- IN THE LAST SECTION WE SAW how second order differential equations naturally appear in the derivations for simple oscillating systems. In this section we will look at more general second order linear differential equations.
- 2.2: Constant Coefficient Equations
- THE SIMPLEST SECOND ORDER DIFFERENTIAL EQUATIONS are those with constant coefficients.
- 2.3: Simple Harmonic Oscillators
- THE NEXT PHYSICAL PROBLEM OF INTEREST is that of simple harmonic motion. Such motion comes up in many places in physics and provides a generic first approximation to models of oscillatory motion. This is the beginning of a major thread running throughout this course. You have seen simple harmonic motion in your introductory physics class. We will review SHM (or SHO in some texts) by looking at springs, pendula (the plural of m pendulum), and simple circuits.
- 2.4: Forced Systems
- ALL OF THE SYSTEMS PRESENTED at the beginning of the last section exhibit the same general behavior when a damping term is present. An additional term can be added that might cause even more complicated behavior. In the case of LRC circuits, we have seen that the voltage source makes the system nonhomogeneous. It provides what is called a source term.
- 2.5: Cauchy-Euler Equations
- Another class of solvable linear differential equations that is of interest are the Cauchy-Euler type of equations, also referred to in some books as Euler’s equation.