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.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve a logistic equation and interpret its carrying-capacity limit.
- Justify the conclusion "y(t)=1/(1+exp(−t))" using the stated assumptions.
Before you start
Separable ODEs and partial fractions.
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
Solve y′=y(1−y), y(0)=1/2.
Why this math matters
Solve a logistic equation and interpret its carrying-capacity limit. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The equation is an ideal nondimensional model.
- The main initial value lies strictly between zero and one.
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.
Distinguish exact equilibrium from growth toward one
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve y′=y(1−y), y(0)=1/2.
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.
Build the model
dy/[y(1−y)]=dt and 1/[y(1−y)]=1/y+1/(1−y)
Separation is valid along this solution, which stays between zero and one.
Work through the mathematics
log(y/(1−y))=t+C; C=0
Integrating and using the initial value determine the log-odds.
Check the conclusion
y(t)=1/(1+exp(−t))
Substitution verifies the equation and shows increasing convergence to one as t→∞.
The result
y(t)=1/(1+exp(−t))
Substitution verifies the equation and shows increasing convergence to one as t→∞.
Common mistakes to catch
- Separation can lose equilibrium solutions.
- Approaching a limit is not the same as reaching it in finite time.
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
Which constant solutions were excluded by division?
Show a hint
Set the original right side to zero.
Reveal answer and explanation
y=0 and y=1
Both equilibria solve the ODE but cannot be found by dividing by y(1−y).
Practice 2
When does this solution reach 3/4?
Show a hint
Solve the log-odds equation.
Reveal answer and explanation
t=log 3
The odds y/(1−y) equal three.
Take the idea with you
Compare early near-exponential growth with later saturation.
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: Use two initial conditions in a second-order motion model
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