clc7eap1504

# clc7eap1504 - SECTION 15.4 Second-Order Nonhomogeneous...

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SECTION 15.4 Second-Order Nonhomogeneous Linear Equations 1117 15.4 SECTION Second-Order Nonhomogeneous Linear Equations Nonhomogeneous Equations • Method of Undetermined Coefficients • Variation of Parameters Nonhomogeneous Equations In the preceding section, we represented damped oscillations of a spring by the homo- geneous second-order linear equation Free motion This type of oscillation is called free because it is determined solely by the spring and gravity and is free of the action of other external forces. If such a system is also subject to an external periodic force such as caused by vibrations at the oppo- site end of the spring, the motion is called forced, and it is characterized by the nonhomogeneous equation Forced motion In this section, you will study two methods for finding the general solution of a nonhomogeneous linear differential equation. In both methods, the first step is to find the general solution of the corresponding homogeneous equation. General solution of homogeneous equation Having done this, you try to find a particular solution of the nonhomogeneous equation. Particular solution of nonhomogeneous equation By combining these two results, you can conclude that the general solution of the nonhomogeneous equation is as stated in the following theorem. y 5 y h 1 y p , y 5 y p y 5 y h d 2 y dt 2 1 p m 1 dy dt 2 1 k m y 5 a sin bt . a sin bt , d 2 y dt 2 1 p m 1 dy dt 2 1 k m y 5 0. THEOREM 15.6 Solution of Nonhomogeneous Linear Equation Let be a second-order nonhomogeneous linear differential equation. If is a partic- ular solution of this equation and is the general solution of the corresponding homogeneous equation, then is the general solution of the nonhomogeneous equation. y 5 y h 1 y p y h y p y 0 1 ay 9 1 by 5 F s x d S OPHIE GERMAIN (1776–1831) Many of the early contributors to calculus were interested in forming mathematical models for vibrating strings and membranes, oscillating springs, and elasticity. One of these was the French mathematician Sophie Germain, who in 1816 was awarded a prize by the French Academy for a paper entitled “Memoir on the Vibrations of Elastic Plates.” The Granger Collection

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1118 CHAPTER 15 Differential Equations Method of Undetermined Coefficients You already know how to find the solution of a linear homogeneous differential equation. The remainder of this section looks at ways to find the particular solution If in consists of sums or products of or you can find a particular solution by the method of undetermined coefficients. The gist of this method is to guess that the solution is a generalized form of Here are some examples. 1. If choose 2. If choose 3. If choose Then, by substitution, determine the coefficients for the generalized solution. EXAMPLE 1
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clc7eap1504 - SECTION 15.4 Second-Order Nonhomogeneous...

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