Final answer:
To find a particular solution to the given differential equation using the method of variation of parameters, we first need to find the complementary solution and then substitute it into the assumed solution form. The particular solution can be expressed in terms of the unknown functions and the linearly independent solutions to the homogeneous equation.
Step-by-step explanation:
To find a particular solution to the given differential equation using the method of variation of parameters, we first need to find the complementary solution which is the solution to the homogeneous equation x^2y'' - 13xy' + 45y = 0. The complementary solution is given by y_c = c1*x^5 + c2*x^9, where c1 and c2 are arbitrary constants.
Next, we need to find the particular solution by assuming a solution of the form y_p = u1(x)*y1(x) + u2(x)*y2(x), where u1(x) and u2(x) are unknown functions.
After substituting this assumed solution into the differential equation and solving for u1 and u2, we find that u1(x) = x^3*ln(x) and u2(x) = x^7*ln(x). Therefore, the particular solution is y_p = x^3*ln(x)*y1(x) + x^7*ln(x)*y2(x), where y1(x) and y2(x) are the two linearly independent solutions to the homogeneous equation.