Final answer:
The given differential equation is (D^2 + 6D + 10)y = 6e^(-3θ)cosθ. The solution to this equation involves solving the auxiliary equation, which yields complex roots. The general solution to the differential equation is y = e^(-3θ)(Acosθ + Bsinθ).
Step-by-step explanation:
The given differential equation is (D^2 + 6D + 10)y = 6e^(-3θ)cosθ. To find the solution, we need to solve the auxiliary equation m^2 + 6m + 10 = 0.
Using the quadratic formula, we can find the roots of the auxiliary equation: m = (-6 ± √(6^2 - 4*1*10))/(2*1) = -3 ± i.
Since we have complex roots, the general solution to the differential equation is y = e^(-3θ)(Acosθ + Bsinθ), where A and B are constants determined by the initial conditions.