For the differential equation y" - 9y' + 18y = 32e2t (a) State the characteristic polynomial for the differential equation. (b) List the roots of the characteristic polynomial. (c) List y1 and y2. (d) Find the homogeneous solution to the differential equation. In your answer, use A and B to denote the arbitrary constants. yn(t) = (e) Find the non-homogeneous solution to the differential equation. Y (t) = (f) State the general solution. y(t) = (g) Find the particular solution satisfying y(0) = 16 and y'(0) = 37. y(t) =] Note: You can earn partial credit on this problem.For the differential equation 3;\" + 4y' 32y = 128'\" (a) State the characteristic polynomial for the differential equation. (b) List the roots of the characteristic polynomial. (c}tisty1 and y2. U (d) Find the homogeneous solution to the differential equation. In your answer, use A and B to denote the arbitrary constants. ya\") =3 (e) Find t_he nonhomogeneous solution to the differential equation. m) =. i (f) State the general solution. W) = i H (9) Find the particular solution satisfying y(0) = 7 and y'(0) = 31. W) =J ate: You can earn partial credit on this problem. Forthe differential equation y\" 12y'l 363; = 294_t (Estate the characteristic polynomial for the differential equation. (b) List the roots of the characteristic polynomial. (c) List yl and m. (d) Find the homogeneous solution to the differential equation. In your answer, use A and B to denote the arbitrary constants. '9": it) =|:| (e) Find the nonhomogeneous solution to the differential equation. m) =|:| (f) StateLhe general solution. W) =|._,| (g) FindLhe particular solution satisfying 9(1)} 2 1 and y'({l) = 57. W) :|'_J For the differential equation 1/\" + 12y' + 361; = 8661 (Estate the characteristic polynomial for the differential equation. (b_)List the roots of the characteristic polynomial. (gList yl and m. (d) Find the homogeneous solution to the differential equation. In your answer. use A and B to denote the arbitrary constants. yhtt) =l:l (e) Find t_he nonhomogeneous solution to the differential equation. Y(t) 2U (f) Stateihe general solution. W) =|'_'| (g) Findt_he particular solution satisfying gm} : 2 and y'(0} : 10. W} =LJ a. Find a particular solution to the nonhomogeneous differential equation 3;" i 4;)\" + 5y 2 15:r: + e m. y? : |:| help (formulas) b. Find the most general solution to the associated homogeneous differential equation. Use c1 and .32 in your answerto denote arbitrary constants) and enter them as c1 and c2. yr; = |:| help (formulas) c. Find the most general solution to the original nonhomogeneous differential equation. Use (:1 and (:2 in your answer to denote arbitrary constants. y : || help (formulas)