Final answer:
To find the general solution of the differential equation, we first solve the associated homogeneous equation to find the complementary function. Then, we assume a particular solution and solve for the constants. The general solution is the sum of the complementary function and the particular solution.
Step-by-step explanation:
To find the general solution of the given differential equation, we can first find the complementary function by solving the associated homogeneous equation, y'' - y' - 6y = 0. The characteristic equation is r^2 - r - 6 = 0, which factors as (r - 3)(r + 2) = 0. So, the complementary function is y_c(x) = C1e^(3x) + C2e^(-2x), where C1 and C2 are arbitrary constants.
To find a particular solution for the non-homogeneous equation, we can use the method of undetermined coefficients. Let's assume a particular solution of the form y_p(x) = Ax + B + Csin(3x) + Dcos(3x), where A, B, C, and D are constants. Substituting this into the equation, we can equate the coefficients of like terms on both sides.
After finding the particular solution, the general solution of the differential equation is y(x) = y_c(x) + y_p(x), where y_c(x) is the complementary function and y_p(x) is the particular solution.