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MTH 256 – Sec 4.1 Sample Problems 8. Determine whether the following functions are linearly independent. If not, find a linear relation among them. f1 (t) = 2t − 3, f2 (t) = 2t2 + 1, f3 (t) = 3t2 + t. Sol. The Wronskian is 2t − 3 2t2 + 1 3t2 + t 4t 6t + 1 W (f1 , f2 , f3 ) = 2 0 4 6 2t − 3 2t2 + 1 2t − 3 3t2 + t +6 = −4 2 4t 2 6t + 1 = −4 (12t2 − 16t − 3) − (6t2 + 2t) + 6 (8t2 − 12t) − (4t2 + 2) = −24t2 + 72t + 12 + 24t2 − 72t − 12 =0 Thus the functions are linearly dependent. To find the relationship, solve for a and b in the following f1 = af2 + bf3 2t − 3 = 2at2 + a + 3bt2 + bt. Thus a = −3 and b = 2. If the functions had been independent, we could not have found a and b to satisfy the above equation. 15. Verify that the given solutions are linearly independent by determining the Wronskian. xy 000 − y 00 = 0, t > 0; 1, x, x3 . Sol. The Wronskian is 1 x x 3 1 x 2 = 6x. W (y1 , y2 , y3 ) = 0 1 3x = 6x 0 1 0 0 6x Thus W (y1 , y2 , y3 ) 6= 0 for x > 0. The solutions form a fundamental set. 1