7.8 Exponential Models with Differential Equations

The exponential growth/decay equation

If a quantity changes at a rate proportional to its current value, it satisfies dy/dx = ky. Solving this separable differential equation gives y = Ce^(kx), where C is the initial value (y at x=0). If k > 0, y grows exponentially; if k < 0, y decays exponentially.

Recognizing dy/dx = ky

Watch for phrases like 'rate of change proportional to the amount present' — this always means dy/dx = ky. Common contexts: population growth, radioactive decay, Newton's Law of Cooling (a shifted version), and continuously compounded interest. Identify k and the initial condition to write the specific solution.

Solving by separation of variables

Separate: dy/y = k dx. Integrate both sides: ln|y| = kx + C₁. Exponentiate: y = e^(kx+C₁) = Ce^(kx), where C = e^(C₁) is found using the initial condition. Always verify the solution by differentiating and plugging back into the original equation.

Key terms

Differential equation for exponential growth/decay

dy/dx = ky — rate of change proportional to the current value of y.

General solution to dy/dx = ky

y = Ce^(kx), where C is the initial value and k is the growth/decay constant.

How to find C

Use the initial condition (value of y when x = 0); C = y(0).

Sign of k for growth vs. decay

k > 0 means exponential growth; k < 0 means exponential decay.

Method to solve dy/dx = ky

Separate variables (dy/y = k dx), integrate both sides, then exponentiate to solve for y.

Try an AP-style question

A quantity satisfies dy/dt = ky, starts at 100, and doubles every 3 years. What is it at t = 6?

1. 600
2. 200
3. 300
4. 400