Chapter03-page5 - y a y by = f x where a and b are...

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Physics 235 Chapter 3 - 5 - The phase paths will be executed in a clock-wise direction. For example, in the upper right corner of the phase diagram, the velocity is positive. This implies that x must be increasing. The x coordinate will continue to increase until the velocity becomes equal to zero. Figure 3. Phase diagram for a one-dimensional simple-harmonic oscillator Solving Second-Order Differential Equations The second-order differential equations that we have discussed in the previous sections are simple equations that can be solved analytically with little effort. Once we start damping and/or driving forces, the equations become more complicated, and we need to discuss in more detail how we can solve these equations. Second-order differential equations have the following form:
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Unformatted text preview: y + a y + by = f x ( ) where a and b are constants. When f ( x ) = 0, the equation is called a homogeneous equation ; otherwise it is called an inhomogeneous equation . Any solution of this equation can be rewritten as a linear superposition of any two linearly independent solutions of this equation. Any solution will have two parameters that need to be adjusted to match the initial conditions. We start by first looking at the homogeneous equation. Consider the following solution of this equation: y x ( ) = e rx If we substitute this solution into the homogeneous equation we get r 2 + ar + b = In general there are two possible values of r :...
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This note was uploaded on 01/08/2012 for the course PHY 235 taught by Professor Morgan during the Spring '09 term at SUNY Stony Brook.

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