Second order equation with constant coefficients

The equation says: a fixed combination of $y$, its slope $y'$ and its curvature $y''$ must cancel to zero at EVERY $x$. The simplest way for three functions to cancel like that everywhere is for all three to be constant multiples of ONE function, so that the equation only has to balance the multiples. So we look for a function whose derivatives are multiples of itself, and the functions that do that are the exponentials $e^{rx}$. The guess is not a trick pulled from a hat, and it costs nothing: a theorem about linear equations (Lebl, Notes on Diffy Qs, Theorem 2.1.3) says that once two independent solutions $y_1$ and $y_2$ are known, EVERY solution is $C_1 y_1 + C_2 y_2$. So if the guess produces two, there are no others left to find.

Put $y = e^{rx}$ into the equation with $r$ still unknown. Each differentiation simply brings down a factor of $r$, so every term contains the same $e^{rx}$, which can be taken out. What is left is an equation about the NUMBER $r$, with no derivatives in it at all. Calculus has been traded for algebra.

It is a quadratic, and its discriminant decides the whole character of the solution before we compute anything else: positive means two real rates (pure growth or decay), zero means one rate counted twice, negative means a complex pair, and complex rates are how oscillation enters an equation that contains no sines or cosines at all.

The solutions $e^{(\alpha \pm \beta i) x}$ are perfectly good, but they are complex, and the problem is about a real quantity. Euler's formula, $e^{i \theta} = \cos \theta + i \sin \theta$, splits them: $e^{(\alpha + \beta i)x} = e^{\alpha x} \left( \cos \beta x + i \sin \beta x \right)$. Because the equation has REAL coefficients, the real part and the imaginary part of a complex solution are each solutions on their own. So an imaginary part in the rate is not a nuisance: it IS the oscillation.

At $x = 0$ the cosine is $1$, the sine is $0$ and the exponential is $1$, so the first starting value gives $C_1$ with no work. The slope at zero then gives $C_2$.

The check for a differential equation asks nothing on trust: differentiate the answer twice, put everything into the left side, and it must vanish for every $x$. Our software does this exactly, in surd arithmetic, comparing the coefficient of each basic function separately, and refuses to show the page if anything is left over.