Consider the following fourth-order nonhomogeneous differential equation with the given initial conditions: y ( 4) - y(3) - y" - y' - 2y = 8x5 y(0) = y'(0) = y"(0) = y(3) (0) = 0 1. What makes this equation nonhomogeneous? 2. Write the associated homogeneous equation. 3. Find yc by solving the associated homogeneous equation. a. Find the characteristic polynomial. b. Find the possible rational roots by using the Rational Roots Theorem. c. Find all roots of the characteristic polynomial. d. Find yc. 4. Find yp by using the Method of Undetermined Coefficients. a. Find the guess for yp- b. Do the terms in part (a) have anything in common with yc in question 3? c. Find yp, yp', yp , and yp*. d. Substitute the values in part (d) in the given nonhomogeneous equation. e. Collect the coefficients for each term. f. Equate the coefficients and solve the system. (It looks long, but is actually a pretty simple substitution.) g. Find actual yp. 5. Find the general solution y(x). 6. Find the solution to the initial value problem by plugging in the initial conditions to the general solution. a. Find y'(x), y"(x), and y(3)(x). Make sure to calculate for both pieces of the general solution. b. Plug in initial condition and find system of coefficients. c. Solve the system of coefficients. (If you find this problem in the text, the answer in the back is incorrect.) d. Write out general solution y(x) with filled in c1, C2, C3, C4 values