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section%204.1 - Chapter 4. Higher-Order Dierential...

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Chapter 4. Higher-Order Diferential Equations § 4.1 Preliminary Theory–Linear Equations 1. Higher-Order Linear Diferential 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 + a 0 ( x ) y = g ( x ) y ( x 0 )= y 0 ,y ° ( x 0 )= y 1 , ··· ,y ( n 1) ( x 0 )= 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 ) ,a 0 ( x ) and g ( x ) be continuous on an interval I and let a n ( x ) ± =0forevery x in I .I f x = x 0 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
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section%204.1 - Chapter 4. Higher-Order Dierential...

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