1. (7 pts) Families of functions. Find the general solution to the equation y" - y' - 2y = 5 cos x using the steps outlined below. (a) Find the general solution of the associated homogeneous equation y" - y' - 2y = 0 which is often referred to as the homogeneous solution. (b) Try the ansatz Up(x) = Acosa + Bsinx. Why would the arguably more obvious ansatz Up() = A cos x not work? (c) Add the homogeneous solution to your particular solution Up to get the general solution to the inhomogeneous equation. 2. (7 pts) Families of functions. Find the general solution to y" - 6y' + 9y = 5xe2x (a) Find the homogeneous solution. (b) Try the ansatz yp(x) = (Ax+ B)e27. Why would the arguably more obvious ansatz yp(x) = Axe27 not work? (c) State the general solution to the inhomogeneous equation.3. (7 pts) When the RHS is in the nullspace. Find the general solution to y" + 3y' + 2y = 7e-2x (a) Find the homogeneous solution. (b) Try the ansatz Up(x) = Ae-2. Why does this not work? (c) Try the ansatz Up(x) = Axe -2x. (d) State the general solution to the inhomogeneous equation. 4. (6 pts) When the RHS is in the nullspace and there is a repeated root. Solve the IVP, y" - 6y' + 9y = 2est, y(0) = y'(0) = 0 (a) Find the homogeneous solution. (b) Try the ansatz Up(2) = Axe3x. (c) Explain (i) why only one term is sufficient and (ii) why neither Axes nor Aest would have worked. (d) State the general solution to the inhomogeneous equation. Use it to find the particular solution that solves the IC given above