Final answer:
An example of a non-homogeneous ODE that cannot be solved using the method of undetermined coefficients but can be solved using reduction of order is provided, along with an explanation of why. The equation y'' + xy = e^x is used as an example.
Step-by-step explanation:
An example of a non-homogeneous ordinary differential equation (ODE) that cannot be solved using the method of undetermined coefficients or annihilators but can be solved using reduction of order is:
y'' + xy = e^x
This equation cannot be solved using the method of undetermined coefficients because the non-homogeneous term e^x is in the form of the homogeneous solutions. The method of annihilators also fails because there is no particular integral that can be found. However, the equation can be solved using reduction of order by assuming a second linearly independent solution y_2(x) = v(x)y_1(x), where y_1(x) is a known solution. Solving for v(x) will give the complete solution.