{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# L61 - Fall 2003 Math 308/501502 6 Theory of Higher-Order...

This preview shows pages 1–2. Sign up to view the full content.

Fall 2003 Math 308/501–502 6 Theory of Higher-Order Linear ODEs 6.1 Basic Theory of Linear ODEs Wed, 24/Sep c 2003, Art Belmonte Summary We give an overview of the theory of n th order linear differential equations; here n 1. Of course, the case where n = 1 was dealt with in Section 2.3, whereas n = 2 is dealt with in Chapter 4. Using summation and matrix notation along with a smattering of linear algebra concepts, we treat the general case once and for all. In the following, I = ( a , b ) is an open real interval. TERMINOLOGY A linear ODE of order n has the form n X k = 0 a k ( x ) y ( k ) ( x ) = b ( x ) . Here the a k depend only on x (the independent variable), not on y (the dependent variable). The equation has constant coefficients if the a k are constants; otherwise, it has variable coefficients . If b ( x ) or g ( x ) is zero on I , then the equation is homogeneous ; otherwise it is nonhomogeneous . With the a k and b continuous on I and a n ( x ) 6= 0 on I , divide to obtain the standard form y ( n ) ( x ) + n X j = 1 p j ( x ) y ( n - j ) ( x ) = g ( x ) or L [ y ] ( x ) = g ( x ) , where L = D n + n j = 1 p j D n - j is called a linear differential operator ; i.e., L " m X i = 1 y i # = m X i = 1 L [ y i ] and L [ cy ] = cL [ y ]. (This follows from properties of differentiation.) DEFINITIONS Let f 1 , . . . , f n be n functions that are differentiable ( n - 1 ) times. The Wronskian matrix is n × n square array of derivatives M = f 1 f 2 · · · f n f 0 1 f 0 2 · · · f 0 n . . . . . . . . . . . . f ( n - 1 ) 1 f ( n - 1 ) 2 · · · f ( n - 1 ) n , Its determinant W is called the Wronskian [determinant]. The m functions f 1 , . . . , f m are linearly dependent on I if there exist constants c 1 , . . . c m , not all zero, such that m X k = 1 c k f k ( x ) = 0 for all x I . Otherwise, the functions are linearly independent . (Note that in the linearly dependent case, one function is a linear combination of the other m - 1 functions.) THEOREMS Existence and Uniqueness Let g ( x ) and p k ( x ) be continuous on I , an interval containing x 0 . For any choice of constants γ k , there exists a unique solution y ( x ) on the entire interval I to the initial value problem y ( n ) ( x ) + n X j = 1 p j ( x ) y ( n - j ) ( x ) = g ( x ), y ( k ) ( x 0 ) = γ k , k = 0 , . . . , n - 1 . Representation of Solutions (Homogeneous Case) Suppose that y 1 , . . . , y n are n solutions on I of y ( n ) ( x ) + n X j = 1 p j ( x ) y ( n - j ) ( x ) = 0 , ( * ) where the p k are continuous on I . If the Wronskian of the y k is nonzero at some point x 0 I , then every solution of (*) on I may be expressed as y ( x ) = n X k = 1 c k y k ( x ) , where the c k are constants.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}