Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Solve a separable growth equation.
- Use an initial value to find the constant.
- Interpret the model's rate and its limitations.
Before you start
Natural logarithms, exponentials, derivatives, and basic antiderivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A fictional biomass model satisfies M′(t) = 0.2M(t), with M(0) = 50 grams and t in hours. What mass does the model predict after five hours?
Why this math matters
Some processes change at a rate proportional to their current amount. More material produces a larger absolute growth rate, so equal time steps do not add equal amounts. A differential equation describes that rule directly. An initial condition anchors one solution among the many functions sharing the same proportional growth pattern.
Set up the model
A useful answer starts with clear assumptions:
- Biomass stays positive and the proportional rate 0.2 per hour remains constant.
- The continuous model ignores resource limits over the five-hour interval.
- This is a mathematical example, not a forecast for an actual organism or process.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
What if the rate depends on the amount already present?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional biomass model satisfies M′(t) = 0.2M(t), with M(0) = 50 grams and t in hours. What mass does the model predict after five hours?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Separate the variables
dM/M = 0.2 dt; ln M = 0.2t + C
Division by M is valid under the positive-mass assumption. Integrating both sides converts the derivative relationship into an equation for the function.
Use the starting mass
M(t) = Ke^0.2t; M(0) = K = 50
Exponentiating absorbs the logarithmic constant into a positive multiplier K. The initial condition identifies the required solution.
Evaluate and verify the rule
M(5) = 50e ≈ 135.91 g; M′ = 10e^0.2t = 0.2M
Substitution gives the mass, and differentiation independently confirms that this function satisfies the original rate equation.
The result
The model predicts about 135.91 grams after five hours.
A continuous proportional rate of 0.2 per hour gives a one-hour growth factor e^0.2 ≈ 1.2214, not exactly 1.20. Unlimited exponential growth is a model assumption that eventually becomes unrealistic for resource-limited systems.
Common mistakes to catch
- Adding 20% of the initial amount each hour creates a different, linear model.
- When separating other equations, check equilibrium solutions before dividing by an expression that could be zero.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
If Q′ = −0.1Q and Q(0) = 100, find Q(10).
Show a hint
A negative proportional rate produces exponential decay.
Reveal answer and explanation
100/e ≈ 36.79
Q(t) = 100e⁻⁰·¹ᵗ, so the exponent is −1 at ten time units.
Practice 2
A model has T′ = −0.5(T − 20) and T(0) = 60. Find T(2).
Show a hint
The excess T − 20 decays exponentially.
Reveal answer and explanation
20 + 40/e ≈ 34.72
T(t) = 20 + 40e⁻⁰·⁵ᵗ; at two time units, the initial excess is multiplied by e⁻¹.
Take the idea with you
Identify whether a process's rate depends on the amount, an excess above equilibrium, or something else before choosing an equation.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Which way does a temperature map increase fastest?
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