Final answer:
A general solution to the differential equation involves solving for the homogeneous part with exponentials and trigonometric functions and finding a particular solution for the inhomogeneous part. The total general solution is the sum of these two parts. Verification is done by substituting back into the original differential equation.
Step-by-step explanation:
To find a general solution for the differential equation (D^2 + 2D + \frac{3}{4}I) y = 3e^x + \frac{9}{2}x, we start by solving the associated homogenous equation (D^2 + 2D + \frac{3}{4})y = 0, whose characteristic equation is r^2 + 2r + \frac{3}{4} = 0. The roots of this equation are complex, so the general solution of the homogenous part, y_h, will involve exponentials and trigonometric functions. We look for particular solutions to the non-homogenous part, y_p, by assuming the form y_p = A(x)e^x + B(x)x for the terms on the right-hand side. After finding y_h and y_p, we combine them to get the general solution y = y_h + y_p, which will solve the original differential equation.
To verify whether a solution is correct, one would substitute y, along with its derivatives, back into the original equation and check if the left-hand side equals the right-hand side. The process of solving such differential equations generally involves techniques such as undetermined coefficients or variation of parameters, and for complex roots, Euler's formula may be used to relate exponentials to trigonometric functions.