Q.Verify that $y = e^{4x}$ is a solution of the differential equation $\dfrac{d^2y}{dx^2} - \dfrac{dy}{dx} - 12y = 0$.

A differential equation typically has infinitely many solutions, all captured at once by its general solution — an expression containing exactly as many independent arbitrary constants as the order of the equation (a first-order equation has one arbitrary constant in its general solution, a second-order equation has two, and so on). For instance y = a sin x + b cos x, carrying two constants, is the general solution of a second-order equation, while y = a sin x alone or y = b cos x alone are each only particular solutions obtained by fixing one of the constants. A particular solution is any solution obtained from the general solution by assigning specific numerical values to its arbitrary constant(s), usually because extra information is given — an initial condition such as 'y = 2 when x = 0', or a point the solution curve must pass through. To VERIFY that a given expression is a solution of a stated differential equation, the expression is differentiated the required number of times, and the resulting derivatives (together with the original expression, and the original relation used again if a constant needs eliminating) are substituted into the differential equation to confirm the equation holds identically. This verification step, and the general/particular distinction, underlie every 'find the particular solution satisfying...' problem in the chapter.