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Consider the non-homogeneous differential equation
y" - 10 y' + 9 y = -3 cos(3t) + 5 sin(3t).
a. Find the roots of the characteristic polynomial, r+ and r-, larger and smaller or equal, respectively.
b. Find the real-valued fundamental solutions to the differential equation associated to the roots in part (a).
c. Use the functions in part (b) to find the general solution to the homogeneous equation, (use the names c and d for any arbitrary constants)
d. Find a particular solution yp to the non-homogeneous equation.
e. Find the solution of the initial value problem y" - 10y' + 9y = -3 cos(3t) +5 sin(3t), y(0) = 2, y'(0) =-2.
Let ubar and vbar be nonzero vectors in 2 or 3 space and let k=||u|| and m=||v||. Show that the vector Wbar=mubar +kvbar bisects the angle between ubar and vbar.
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