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 di erential 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 di erential 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.
