Exponential Growth and Decay Models
dy/dt = ky gives y = y₀eᵏᵗ.
Key idea
dy/dt = ky gives y = y₀eᵏᵗ.
In depth
If a quantity changes at a rate proportional to itself, y′ = ky, then y = y₀e^(kt). k > 0 is growth; k < 0 is decay, with half-life ln 2/|k|. It models populations, radioactivity and compound interest.
Worked example
A substance with half-life 5 years: k = −ln 2/5; after 15 years 1/8 remains.
Formula
Real-world uses
- Modelling bacteria growth, radioactive decay and the value of an investment with continuous interest
Common mistake
Applying it when the growth rate is limited by resources.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Exponential Growth and Decay Models?
dy/dt = ky gives y = y₀eᵏᵗ. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Exponential Growth and Decay Models?
The key formula is y = y₀ e^(kt).
Where is Exponential Growth and Decay Models used in real life?
Real-world examples: Modelling bacteria growth, radioactive decay and the value of an investment with continuous interest.
What should I learn before Exponential Growth and Decay Models?
Make sure you are comfortable with First-Order Linear Equations, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Exponential Growth and Decay Models?
Applying it when the growth rate is limited by resources.
How can I practise Exponential Growth and Decay Models?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Exponential Growth and Decay Models?
Move on to Logistic Growth next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- First-Order Linear Equations – Use an integrating factor.
- Logistic Growth – Growth limited by carrying capacity.