Find the complementary function to the second order differential equation d^2y/dx^2 - 5dy/dx + 6x = x^2

Use the auxiliary equation k²-5k+6=0. Solving this gives roots k=2 and k=3, which are real and distinct roots. This means that the complementary function is of the form y=Ae^(k₁x)+Be^(k₂x), where k₁ and k₂ are roots of the auxiliary equation and A and B are real constants. Therefore the complementary function for this differential equation is y=Ae^2t+Be^3t.