What Is a Differential Equation?

A differential equation relates a function to its derivatives: $F(x, y, y', y'', \ldots) = 0$. Its order is the highest derivative present; it is linear when $y$ and its derivatives appear only to the first power, with no products between them.

A solution is any function that satisfies the equation identically — verify by substituting in and differentiating, never by solving. General solutions contain arbitrary constants; initial conditions pin them to a particular solution.

Verify $y = e^{2x}$ solves $y'' - 4y = 0$: $y' = 2e^{2x}$, $y'' = 4e^{2x}$, so $y'' - 4y = 4e^{2x} - 4e^{2x} = 0$ for every $x$. The equation is second order and linear.