dy/dt = ky says the rate of change is proportional to the amount present. It is the single most common differential equation in applications.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The solution
Separating and integrating gives y = y₀e^(kt), where y₀ is the amount at time zero.
The sign of k
A positive k grows and a negative k decays. Nothing else about the shape changes.
Where it appears
Population growth, radioactive decay, continuous interest and Newton cooling all reduce to this equation.
Half-life and doubling time
These are properties of k alone, independent of the starting amount. A sample halves in the same time regardless of size.
Where the model fails
Unlimited exponential growth is unrealistic. A logistic model, which BC covers, adds the carrying capacity this one ignores.
When the rate is proportional to the amount
dy/dt = ky says the growth rate is proportional to the current amount. Separating and integrating gives y = Ce^(kt). That single differential equation generates every exponential model.
The sign of k
Positive k gives growth, negative gives decay. Half-life and doubling time both come from solving for when the amount reaches half or twice its start, and neither depends on the starting quantity.
Where the proportionality holds
Radioactive decay, compound interest, unconstrained population growth, and drug elimination. The common feature is that the change depends only on how much is currently present.
And where it stops holding
Populations run out of food and epidemics run out of hosts, so real growth curves flatten. The logistic equation models that ceiling. Knowing where the exponential model expires is part of using it honestly.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 1)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: Watch an Example
One step at a time.
Watch Kofi Solve a Growth Equation
A population satisfies dP/dt = 0.05P with P(0) = 200.
- Step 1
The rate is proportional to the amount, so the solution is exponential.
Step 4: Your Turn
Practice makes it stick.
The Constant
Problem 1 of 2
dP/dt = 0.05P with P(0) = 200. What is P₀?
The Decay
Problem 2 of 2
A sample of 80 g halves every hour. How many grams remain after 3 hours?
g
Growth and Decay
1 of 8
dy/dt = ky with k = 0.03. Growing or decaying? 1 growing, 2 decaying.
2 of 8
dy/dt = ky with k = −0.2. Growing or decaying? 1 or 2?
3 of 8
A sample of 160 g halves every hour. Grams after 4 hours?
4 of 8
A population doubles every 5 years. By what factor in 15 years?
5 of 8
Does half-life depend on the starting amount? 1 yes, 0 no.
6 of 8
y = y₀e^(kt) at t = 0. What is y?
7 of 8
Which situations follow dy/dt = ky?
8 of 8
A sample of 64 g halves every hour. Grams after 3 hours?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A sample of 100 g halves every hour. How many grams remain after 2 hours?
Question 2 of 2
What does dy/dt = ky say?
What You Learned
- dy/dt = ky has solution y = y₀e^(kt).
- Positive k grows and negative k decays.
- Half-life and doubling time depend only on k, never on the starting amount.