Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive the step-size stability condition for forward Euler.
- Justify the conclusion "0<h<0.4 gives decay; 0<h≤0.2 avoids sign alternation" using the stated assumptions.
Before you start
Euler's method and geometric sequences.
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
Apply forward Euler to y′=−5y and determine when errors decay.
Why this math matters
Derive the step-size stability condition for forward Euler. 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 timestep h is positive and constant.
- The test equation is solved without roundoff for this analysis.
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.
Check whether a numerical decay method actually decays
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Apply forward Euler to y′=−5y and determine when errors decay.
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
yₙ₊₁=yₙ+h(−5yₙ)=(1−5h)yₙ
Each step multiplies the state and its linear perturbation by one factor.
Work through the mathematics
|1−5h|<1
Decay requires the amplification factor to lie strictly inside the unit circle.
Check the conclusion
0<h<0.4 gives decay; 0<h≤0.2 avoids sign alternation
A stable method can still produce oscillating approximations when the factor is negative.
The result
0<h<0.4 gives decay; 0<h≤0.2 avoids sign alternation
A stable method can still produce oscillating approximations when the factor is negative.
Common mistakes to catch
- A method's consistency does not guarantee stability for a chosen step.
- Qualitative decay can fail even when the differential equation is simple.
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
What happens at h=0.5?
Show a hint
Compute the amplification factor.
Reveal answer and explanation
Magnitudes grow by factor 1.5 per step with alternating sign
The factor is −1.5, although the true solution decays.
Practice 2
What happens at h=0.4?
Show a hint
The factor equals −1.
Reveal answer and explanation
No decay; values alternate
This is the stability boundary, not strict decay.
Take the idea with you
Select a timestep based on the fastest decay scale in a model.
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 a midpoint slope to improve one numerical step
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