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8. Assume that y, = c' is a solution of y' 1' -- 2y - 2(1 +2-+ Use 5 above to find a particular solution of y" + y' - 2y = 6(1 + x -'). Answer: y, = 3x2. Verify the correctness of this result. 9. Assume that yo, = 1 + z is a solution of -yty=1, and 1pz = e"* is a solution of v" - y'ty = 3ex. Use 6 above to find a particular solution of y" - y' ty = 1+ 30". Answer: yp = 1 + + + ex. Verify the correctness of this result. 10. Assume that yp = (7ocosx - 7', sin ) + i(7, sin z + 7, cos z) is a solu- tion of y" - 37' + 2y = ez = cosx + isinz. Use 7 above to find a particular solution of (a) y" - 3y' + 2y ~ cos x, (b) y" - 3y' + 2y = sinc. Ans. (@) y(@) = 17cost - To sing. (D) V(@) = 16 sing- 16 cost. Verify the correctness of each of these results.Il. Prove that two functions are linearly dependent if one is a constant multiple of the other. 12. Assume f1, f2, f3 are three linearly independent functions. Show that the addition to the set of one of these functions, say f1, makes the new set linearly dependent. 13. Prove that a set of functions f1, f2, . . . , fx, is linearly dependent if two functions of the set are the same. 14. Prove that a set of functions, fo, f2, . . ., fa, is linearly dependent, if a subset, i.e., if a part of the set, is linearly dependent