- The functions f1(x) = 5x; f2(x) = x
2 + 3x, and f3(x) = 2x
2 + x do not form a linearly
independent set of functions. Verify this by writing one of the given functions as a linear
combination of the other two functions. (2 pts.) - Show that the functions f1(x) = e
4x
and f2(x) = e
2x
form a linearly independent set of
functions. (2 pts.) - Suppose that the auxiliary equation of a fth-order homogeneous linear dierential equation
with constant coecients has roots m1 = 1; m2 = 1; m3 = 4; m4 = 2 + i, and
m5 = 2 i. What is the solution to the dierential equation? (2 pts.) - Solve y
000 y
00 9y
0 + 9y = 0. (4 pts.) - Use the method of undetermined coecients to solve y
00 6y
0 + 8y = 2x+ 1 + 4e
2x
. (5 pts.) - Use variation of parameters to solve y
00 4y
0 + 3y = e
5x
. (5 pts.
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