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1. Prove that if p = q, the functions eat and e are linearly independent. Hint. Follow the method used in proving the linear independence of the functions in Example 19.13-2. 2. Prove that the functions cox and re" are linearly independent. Hint. Make use of Example 19.13-2 with p = 0, q = 1, and the fact that er # 0 for all z. 3. Prove that the functions sin z, 0, cos r are linearly dependent. 4. Prove that the functions 3es and -2e2z are linearly dependent.It is extremely important that you prove the statements in Exercises 5 to below. Follow the method used to prove statements 2 and 3 of Theorem 9.3. 5. If yp is a solution of 19.5) f(xly(+ .. .+ ()y' t foxy = Q(x), then Ay, is a solution of (19.5) with Q(x) replaced by AQ(z).6. Principle of Superposition. (Also see Comment 24.25.) If y,, is a solution of (19.5) with Q(x) replaced by Qi(r) and yp, is a solution of (19.5) with Q(x) replaced by Q2(t), then yp = yp, + y>, is a solution of f ( = ) y( + . . . + fi(xy' + folz)y = QI(x) + Q2(r). 7. If ya(x) - u(x) + in(x) is a solution of (19.51) f . (x )y() + . . . + f(x)y + fo(x)y = R(z) + is(I), where fo(I), . . ., f=(x) are real functions of z, then (a) the real part of yp, i.e., u(z), is a solution of f(ry() + .. . + (x)y' + fo(xy = R(I), (b) the imaginary part of y,, i.e., *(x), is a solution of f.(x) y()+ . .. + (x)y' + fo(x)y = S(x). Hint. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal