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Homog_DEs_Results - 14 Homogeneous Linear Equations The Big...

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14 Homogeneous Linear Equations — The Big Theorems Let us continue the discussion we were having at the end of section 12.3 regarding the general solution to any given homogeneous linear differential equation. By then we had seen that any linear combination of particular solutions, y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + · · · + c M y M ( x ) , is another solution to that homogeneous differential equation. In fact, we were even beginning to suspect that this expression could be used as a general solution to the differential equation provided the y k ’s were suitably chosen. In particular, we suspected that the general solution to any second-order, homogeneous linear differential equation can be written y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) were c 1 and c 2 are arbitrary constants, and y 1 and y 2 are any two solutions that are not constant multiples of each other. These suspicions should have been reinforced in the last chapter in which general solutions were obtained via reduction of order. In the examples and exercises, you should have noticed that the solutions obtained to the given homogeneous differential equations could all be written as just described. It is time to confirm these suspicions, and to formally state the corresponding results. These results will not be of merely academic interest. We will use them for much of the rest of this text. For practical reasons, we will split our discussion between this and the next chapter. This chapter will contain the statements of the most important theorems regarding the solutions to homogeneous linear differential equations, along with a little discussion to convince you that these theorems have a reasonable chance of being true. More convincing (and lengthier) analysis will be carried out in the next chapter. 14.1 Preliminaries and a Little Review We are discussing general homogeneous linear differential equations. If the equation is of second order, it will be written as ay ′′ + by + cy = 0 . 297
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298 Homogeneous Linear Equations — The Big Theorems More generally, it will be written as a 0 y ( N ) + a 1 y ( N 1 ) + · · · + a N 2 y ′′ + a N 1 y + a N y = 0 where N , the order, is some positive integer. The coefficients — a , b and c in the second order case, and the a k ’s in the more general case — will be assumed to be continuous functions over some open interval I , and the first coefficient — a or a 0 — will be assumed to be nonzero at every point in that interval. Recall the “principle of superposition”: If { y 1 , y 2 , . . . , y K } is a set of particular solutions over I to a given homogeneous linear equation, then any linear combination of these solutions, y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + · · · + c K y K ( x ) for all x in I , is also a solution over I to the the given differential equation. Also recall that this set of y ’s is called a fundamental set of solutions (over I ) for the given homogeneous differential equation if and only if both of the following hold: 1. The set is linearly independent over I (i.e., none of the y k ’s is a linear combination of the others over I ).
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