Download MTH 256 – Sec 4.1 Sample Problems

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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.
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