Final answer:
The solution involves finding the complementary solution from the characteristic equation and then determining the particular solution using the method of undetermined coefficients to obtain the general solution.
Step-by-step explanation:
The student is asking for a solution to a second-order linear differential equation using the method of undetermined coefficients. To solve this equation, one must first find the characteristic equation and its roots to determine the complementary solution. Once the complementary solution is found, a particular solution is assumed based on the form of the non-homogeneous term (4x² + 19). This assumed particular solution is then substituted back into the differential equation to determine the undetermined coefficients. Finally, adding the complementary and particular solutions gives the general solution to the differential equation.