Why Is the Solution of a Linear Nonhomogeneous Constant-Coefficient Differential Equation the Sum of a Particular and Homogeneous Solution?
My textbook gives the following proof:
Let $x_c(t)$ be a solution of the homogeneous equation and $x_p(t)$ be a solution of the particular equation.
$a\dfrac{d^2x}{dt^2} + b\dfrac{dx}{dt} + cx = a\left( \dfrac{d^2x_c}{dt^2} + \dfrac{d^2x_p}{dt^2} \right) + b\left( \dfrac{dx_c}{dt} + \dfrac{dx_p}{dt} \right) + c(x_c + x_p)$
$= \left( a\dfrac{d^2x_c}{dt^2} + b\dfrac{dx_c}{dt} + cx_c \right) + \left( a\dfrac{d^2x_p}{dt^2} + b\dfrac{dx_p}{dt} + cx_p \right)$
$= 0 + f(t) = f(t) $
I don't see how this proves that a homogeneous solution and a particular solution are required to find the solution of a linear non homogeneous constant-coefficient differential equation?
In fact, the proof shows that we end up with $= 0 + f(t) = f(t) $ where $f(t)$ is what we were seeking all along. Therefore, since $\left( a\dfrac{d^2x_c}{dt^2} + b\dfrac{dx_c}{dt} + cx_c \right) = 0$, what was the point of the solution to the homogeneous equation? It seems that the homogeneous solution was redundant from the beginning, since only the particular solution, $\left( a\dfrac{d^2x_p}{dt^2} + b\dfrac{dx_p}{dt} + cx_p \right)$, contributed towards the solution of the linear non homogeneous constant-coefficient differential equation ($\left( a\dfrac{d^2x_p}{dt^2} + b\dfrac{dx_p}{dt} + cx_p \right) = f(t)$).
I would greatly appreciate it if people could please take the time to clarify concept.
EDIT
I've received 3 answers; none of which address my question or do it sufficiently. My question is very clear: How does the above calculation show that both a particular solution and a homogeneous solution are required for a general solution? I then mentioned that the homogeneous solutions ends up equating to $0$ and the particular solution equates to $f(x)$; in other words, it seems that the particular solution was the only thing required to get the general solution $f(x)$, which seemingly makes the homogeneous solution redundant? What am I misunderstanding?
Given the poor response it has received, I am going to request that a moderator delete this question.