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University · Differential equations

Follow exponential growth and decay

Connect an amount to its rate of change through the simplest proportional differential equation.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Follow exponential growth and decay. Elapsed time: 0 time units. Amount y: 1.5. Rate dy/dt: 0.45 amount/time. Behavior: Exponential growthAmount y · dy/dt = ky012340816tDashed navy: instantaneous tangent
The rate constant is fixed and y₀ is positive. This is an exact scalar ODE solution, not a numerical simulation or a fluid-flow PDE. The graph spans t from 0 to 4 and y from 0 to 16 in arbitrary units. No resource limit, noise, or carrying capacity is included.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Elapsed time
0 time units
Amount y
1.5
Rate dy/dt
0.45 amount/time
Behavior
Exponential growth

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

In the model y′ = ky, the rate of change is proportional to the amount currently present. A positive k produces growth, a negative k produces decay, and k = 0 keeps the amount constant. The exact solution y = y₀e^(kt) lets us compare every point with its instantaneous rate.

A relationship to keep

dy/dt = ky; y(t) = y₀e^(kt)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Start with an amount

    At t = 0 the curve begins at y₀. Raising y₀ multiplies the entire solution by the same factor while the proportional rate k stays unchanged.

  2. STEP 2

    Compare amount and rate

    The moving amber point follows the exact solution. Its short dashed tangent shows local direction; the numerical rate ky below the graph distinguishes the amount from how quickly it is changing.

  3. STEP 3

    Change the sign of k

    Choose a negative k and replay. The solution remains positive but approaches zero as time increases. At k = 0 the rate is zero and the graph is horizontal. The four displayed time units are only a finite window.

Your turn to explain

Make a prediction. Test your reasoning.

If y₀ = 2 and k = −0.5, what are y(2) and y′(2)?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

y(2) = 2e⁻¹ ≈ 0.736 amount units. Its instantaneous rate is −e⁻¹ ≈ −0.368 amount units per time unit.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Advanced · Calculus

What if the rate depends on the amount already present?

Solve an exponential growth equation, apply an initial condition, and distinguish continuous rate from a finite percentage change.

Undergraduate · Calculus

Solve a linear ODE by building a product derivative

Choose an integrating factor that combines two terms into one derivative.