Constant-Coefficient Homogeneous Equations

Solving ay″ + by′ + cy = 0 sounds hard — until you ask: what function is its own derivative, over and over, just scaled? The exponential e^{rx}. Differentiate it and you get r·e^{rx}, again and again. So if you GUESS y = e^{rx}, every derivative is just a power of r times the same exponential — and the whole differential equation collapses into ordinary algebra.