7.7 Exponential Models with Differential Equations

An exponential model comes from the statement “the rate of change of a quantity is proportional to the size of the quantity,” which is written as Here, y is the quantity changing, t is usually time, and k is the constant of proportionality. If k>0, the quantity grows because the rate of change has the same sign as y; if k<0, the quantity decays because the rate of change is negative when y is positive. The general solution is where C is an arbitrary constant determined by the initial condition. This means there are infinitely many possible exponential curves solving the same differential equation, and the specific one you use depends on the starting value. If the initial condition is when t=0, then so the particular solution becomes In this form, C or y_0 is the initial amount, k controls how fast the change happens, and describes repeated continuous growth or decay over time. A larger positive k means faster growth, while a more negative k means faster decay. This is the standard AP model for exponential growth and decay in context.