I was trying to understand where the general solution for a linear second-order nonhomogenous differential equation comes from and I don't understand how/why $$Y1 - Y2$$ can turn into $$y(x) - Y_p(x)$$ where $y(x)$ is the general solution and $Y_p(x)$ is the particular solution.
How can we just say one is a general solution and one is a particular solution?
How will the particular solution always be equal to the general solution if its particular?
If this is a common question, guidance on what to search up to find the answer would be appreciated.
$y$ and $Y_p$ are both solutions to the non-homogeneous DE, so $y-Y_p$ is a solution to the homogeneous DE, and since the vector space of solutions to the homogeneous DE is spanned by $y_1$ and $y_2$, we must have that $y-Y_p$ is a linear combination of $y_1$ and $y_2$. In other words, there are $c_i$ such that
$$y - Y_p = c_1 y_1 + c_2 y_2$$
i.e.
$$y = c_1 y_1 + c_2 y_2 + Y_p$$