Final answer:
To solve the given differential equations using the D-operator method, we find the characteristic equation and solve for the values of m. Then, we use the general solutions to determine the specific solutions for each equation.
Step-by-step explanation:
To solve the given differential equations using the D-operator method, we will use the fact that the D-operator represents differentiation. Let's solve each equation step-by-step:
Equation A:
To solve Y'' - 2y' + Y = Xexsinx, we first find the characteristic equation by substituting Y = e^(mx) into the equation:
(m^2 - 2m + 1)e^(mx) = 0
Simplifying, we get m - 1 = 0, so m = 1. This gives us the solution Y = c1e^x + c2xe^x.
Equation B:
To solve Y'' - 4y' + 3y = sin^3(x)cos^2(x), we again find the characteristic equation and solve for m. This time, we get m^2 - 4m + 3 = 0, which factors as (m - 1)(m - 3) = 0. So, m = 1 or m = 3. The general solution is Y = c1e^x + c2e^(3x).