In the last chapter, we studied linear homogeneous constant coefficient (LHCC) equations—problems where the right-hand side was always zero. Their solutions were obtained entirely from exponential functions determined by the characteristic equation.
Now we take the next step: linear nonhomogeneous constant coefficient (LNCC) equations. These look similar, but with one crucial change—there’s a non-zero function on the right-hand side, called the forcing function. This function represents whatever drives or influences the system: an external force, an input, a signal, or another effect.
Solving these equations involves blending two ideas. First, we find the homogeneous solution, which behaves just like the solutions from the previous chapter. Then we construct a particular solution that accounts for the forcing function. Add them together, and you have the general solution.
In this chapter, we’ll learn how to recognize that structure and then develop a powerful tool—the Method of Undetermined Coefficients—to systematically build the particular solution. This will open the door to solving a huge range of real-world problems.
The general solution to a nonhomogeneous differential equation is \(y = y_h + y_p\text{,}\) where \(y_h\) solves the homogeneous equation and \(y_p\) accounts for the forcing function.
The Method of Undetermined Coefficients provides a systematic way to find \(y_p\) when \(f(x)\) consists of polynomials, exponentials, sines, cosines, or their combinations.
