# Chapter 6 - CHAPTER 6 Linear Dierential Systems 6.1...

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CHAPTER 6 Linear Diﬀerential Systems 6.1 Higher-Order Linear Diﬀerential Equations This section is a continuation of Chapter 3. As you will see, all of the “theory” that we developed for second-order linear diﬀerential equations carries over, essentially verbatim, to linear diﬀerential equations of order greater than two. Recall that a Frst order, linear diﬀerential equation is an equation which can be written in the form y ± + p ( x ) y = q ( x ) where p and q are continuous functions on some interval I . A second order, linear diﬀerential equation has an analogous form. y ±± + p ( x ) y ± + q ( x ) y = f ( x ) where p, q , and f are continuous functions on some interval I . In general, an n th -order linear diﬀerential equation is an equation that can be written in the form y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + ··· + p 1 ( x ) y ± + p 0 ( x ) y = f ( x ) (L) where p 0 ,p 1 ,. . . n - 1 , and f are continuous functions on some interval I . As before, the functions p 0 1 . . n - 1 are called the coeﬃcients , and f is called the forcing function or the nonhomogeneous term . Equation (L) is homogeneous if the function f on the right side is 0 for all x I .In this case, equation (L) becomes y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + + p 1 ( x ) y ± + p 0 ( x ) y = 0 (H) Equation (L) is nonhomogeneous if f is not the zero function on I , i.e., (L) is nonhomoge- neous if f ( x ) ± = 0 for some x I . As in the case of second order linear equations, almost all of our attention will be focused on homogeneous equations. Remarks on “Linear.” Intuitively, an n th -order diﬀerential equation is linear if y and its derivatives appear in the equation with exponent 1 only, and there are no so-called ”cross-product” terms, yy ± ,yy ±± ,y ± y ±± , etc. If we set L [ y ]= y ( n ) + p n - 1 y ( n - 1) + + p 1 ( x ) y ± + p 0 ( x ) y , then we can view L as an “operator” that transforms an n -times diﬀerentiable function y = y ( x ) into the continuous function L [ y ( x )] = y ( n ) ( x )+ p n - 1 ( x ) y ( n - 1) ( x + p 1 ( x ) y ± ( x p 0 ( x ) y ( x ) . 251

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It is easy to check that, for any two n -times diﬀerentiable functions y 1 ( x ) and y 2 ( x ), L [ y 1 ( x )+ y 2 ( x )] = L [ y 1 ( x )] + L [ y 2 ( x )] and, for any n -times diﬀerentiable function y and any constant c , L [ cy ( x )] = cL [ y ( x )] . Therefore, as introduced in Section 2.1, L is a linear diﬀerential operator. This is the real reason that equation (L) is said to be a linear diﬀerential equation. ± THEOREM 1. (Existence and Uniqueness Theorem) Given the n th - order linear equation (L). Let a be any point on the interval I , and let α 0 1 ,. . . n - 1 be any n real numbers. Then the initial-value problem y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + ··· + p 1 ( x ) y ± + p 0 ( x ) y = f ( x ); y ( a )= α 0 ,y ± ( a α 1 . . ( n - 1) ( a α n - 1 has a unique solution. Remark: We can solve any Frst order linear diﬀerential equation, see Section 2.1. In contrast, there is no general method for solving second or higher order linear diﬀerential equations . However, as we saw in our study of second order equations, there are methods for solving certain special types of higher order linear equations and we shall look at these later in this section. ± Homogeneous Equations y ( n ) + p n - 1 ( x ) y ( n - 1) + p n - 2 ( x ) y ( n - 2) + + p 1 ( x ) y ± + p 0 ( x ) y =0 . (H)
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Chapter 6 - CHAPTER 6 Linear Dierential Systems 6.1...

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