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# ch03_3 - Ch 3.3 Linear Independence and the Wronskian Two...

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Ch 3.3: Linear Independence and the Wronskian Two functions f and g are linearly dependent if there exist constants c 1 and c 2 , not both zero, such that for all t in I . Note that this reduces to determining whether f and g are multiples of each other. If the only solution to this equation is c 1 = c 2 = 0, then f and g are linearly independent . For example, let f ( x ) = sin2 x and g ( x ) = sin x cos x , and consider the linear combination This equation is satisfied if we choose c 1 = 1 , c 2 = -2, and hence f and g are linearly dependent. 0 ) ( ) ( 2 1 = + t g c t f c 0 cos sin 2 sin 2 1 = + x x c x c

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Solutions of 2 x 2 Systems of Equations When solving for c 1 and c 2 , it can be shown that Note that if a = b = 0, then the only solution to this system of equations is c 1 = c 2 = 0, provided D 0. b y c y c a x c x c = + = + 2 2 1 1 2 2 1 1 2 1 2 1 1 1 2 1 2 1 1 1 2 2 2 2 1 2 1 2 2 1 where , , y y x x D D bx ay x y y x bx ay c D bx ay x y y x bx ay c = + - = - + - = - = - - =
Example 1: Linear Independence (1 of 2) Show that the following two functions are linearly

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ch03_3 - Ch 3.3 Linear Independence and the Wronskian Two...

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