I'm an A-Level maths student, and earlier in the year we learnt about second order differential equations in the form $ A\frac{d^2y}{dx^2} + B \frac{dy}{dx} + Cy = 0$ and $ A\frac{d^2y}{dx^2} + B \frac{dy}{dx} + Cy = f(x)$
For the first form, we let $y = e^{mx} $ (1), then $\frac{dy}{dx} = me^{mx}$ and $\frac{d^2y}{dx^2} = m^2e^{mx} $. This means $Am^2 + Bm + C = 0$ (2). Substituting the solutions into (1) gives us $y=e^{m_0 x}$ and $y=e^{m_1 x}$, where $m_0$ and $m_1$ are the solutions to (2). From this, we somehow conclude that $y = c_1e^{m_0 x} + c_2e^{m_1 x}$
But what I don't understand is, where to the constants come from, and why are the two solutions summed?
Regarding the second form of the equation, the above method breaks down because we cannot solve the quadratic. Why can we ignore the $f(x)$ at this stage?
When $f(x)$ is present, we come up with a different solution (with no constants) called the particular integral. Where does the particular integral come from, and why is that also added to the other solution we obtained by ignoring the $f(x)$?