4.1 Preliminart Theory-Linear Equations- 55 -Chapter 4. Higher-Order Di f erential Equations § 4.1 Preliminary Theory–Linear Equations 1. Higher-Order Linear Di f erential equations: n-th order initial-value problem. a n ( x ) d n y dx n + a n − 1 ( x ) d n − 1 y dx n − 1 + ··· + a 1 ( x ) dy dx + a0 ( x ) y = g ( x ) y ( x0 )= y0 ,y ° ( x0 )= y 1 , ··· ,y ( n − 1) ( x0 )= y n − 1 . (1) If g ( x ) = 0 then the DE is said to be homogeneous. Otherwise, the DE is said to be nonhomogeneous. 2. Existence and Uniqueness: Let a n ( x ) ,a n − 1 ( x ) , ··· ,a 1 ( x ) ,a0 ( x ) and g ( x ) be continuous on an interval I and let a n ( x ) ± =0forevery x in I . If x = x0 is any point in the interval, then a solution y ( x ) of the initial value problem (1) exists on the interval and is unique. 3. Superposition Principle-Homogeneous DE: Let y 1 ,y 2 , ··· ,y k be solutions of the homogeneous n th-order DE on an interval I . Then the linear combination y = c 1 y 1 + c 2 y 2 + ··· + c k y k where c 1 ,c 2 , ··· ,c k are arbitrary constants, is also a solution on the interval.
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This note was uploaded on 01/05/2011 for the course MATH 3310 taught by Professor Dr.du during the Fall '08 term at Kennesaw.