Exponential Models with Differential Equations

Think about why this must be exponential before you ever solve anything. If dy/dt = ky, then the bigger y gets, the faster it changes, and the faster it changes, the bigger it gets. That feedback loop is exactly what exponential growth or decay means: constant percentage change per unit time, not constant amount of change. A population growing at 4 percent per year adds more actual people each year as it gets bigger, because 4 percent of a bigger number is a bigger number.

To solve dy/dt = ky, separate the variables: dy/y = k dt. Integrate both sides to get ln|y| = kt + C. Exponentiate both sides to undo the natural log: y = e^kt+C = e^C e^kt. Rename the constant e^C as y0, the value of y when t = 0, and you get the general solution y = y0 e^kt. The sign of k decides everything: positive k means growth, negative k means decay, and y0 sets the starting amount.

This is the payoff of the whole unit. Slope fields let you sketch what a solution to dy/dt = ky looks like without solving anything, and separation of variables is the technique that turns that picture into an exact formula. Exponential models are the most common application you will actually use that technique for: population growth, radioactive decay, cooling objects, compound interest. Once you see y = y0 e^kt as the solution to a rate-proportional-to-amount setup, you can recognize it in word problems instantly, even before you touch an integral sign.