Consider the DE d²y dy - 18- +81y = x dx² dx which is linear with constant coefficients. First we will work on solving the corresponding homogeneous equation. The auxiliary equation (using m as your variable) is = 0 which has root Because this is a repeated root, we don't have much choice but to use the exponential function corresponding to this root: Y2 = to do reduction of order. Then (using the prime notation for the derivatives) = 3½ = So, plugging y2 into the left side of the differential equation, and reducing, we get y — 18, + 81y2 = So now our equation is eu" = x. To solve for u we need only integrate ace first constant of integration and b as the second we get 9x twice, using a as our u = Therefore y2 = , the general solution. We knew from the beginning that ex was a solution. We have worked out is that re⁹ is another solution to the homogeneous equation, which is generally the case when we have multiple roots. Then 2+9x is the particular solution to the nonhomogeneous equation, and the general solution we derived is pieced together using superposition. 729