Differential equations introduction

A differential equation relates a function to its own rate of change, and a slope field visualizes possible solutions by drawing a short line segment at each point showing that point's slope, without solving the equation algebraically. The simplest solvable kind, dy/dt = ky, describes exponential growth or decay and solves to y = C times e^(kt): for dy/dt = 0.5y with y(0) = 10, C = 10, so y = 10e^(0.5t), meaning at t = 2, y = 10e is about 27.2.

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