Undergraduate · Euler stability · 630 of 650
Euler stability · Decay rate a=0.5; Step size h=1.3
Use n=6 updates, corresponding to time t=nh=7.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=0.5; Step size h=1.3.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Apply explicit Euler to y′=−0.5y, y₀=1, with fixed step h=1.3. Use n=6 updates, corresponding to time t=nh=7.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
One update multiplies by 1−ah. After six updates the result is (1−ah)⁶; the exact value is exp(−a·6h).
Work through the reasoning
Step 1
Identify the model and target
Apply explicit Euler to y′=−0.5y, y₀=1, with fixed step h=1.3. Use n=6 updates, corresponding to time t=nh=7.8. The governing relation is y′=−ay; yₙ=(1−ah)ⁿ; exact y(nh)=e^(−anh). One update multiplies by 1−ah. After six updates the result is (1−ah)⁶; the exact value is exp(−a·6h).
Step 2
Substitute and calculate
One Euler update multiplies by 1−(0.5)(1.3)=0.35. After n=6 steps, yₙ=(0.35)^6=0.001838; the exact solution at time nh=7.8 is exp(−0.5(7.8))=0.020242.
Step 3
Check the mathematical meaning
The signed factor is 0.35. Its magnitude 0.35 is below one, so repeated numerical magnitudes decay. The exact solution always decays for a>0. Euler step: 6; Computed value: 0.002; Exact value: 0.02; Amplification |1−ah|: 0.35: decays. Decimal values are rounded, so use unrounded intermediate values.
The answer
One Euler update multiplies by 1−(0.5)(1.3)=0.35. After n=6 steps, yₙ=(0.35)^6=0.001838; the exact solution at time nh=7.8 is exp(−0.5(7.8))=0.020242. The signed factor is 0.35. Its magnitude 0.35 is below one, so repeated numerical magnitudes decay. The exact solution always decays for a>0. Animation check: Euler step: 6; Computed value: 0.002; Exact value: 0.02; Amplification |1−ah|: 0.35: decays. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
The mathematical idea
An exact decaying differential equation can produce a growing numerical approximation when the time step is too large. Euler's amplification factor is 1−ah, and its magnitude determines stability for this test equation. Negative factors create alternating signs that the positive exact solution never has. Apply explicit Euler to y′=−0.5y, y₀=1, with fixed step h=1.3. Use n=6 updates, corresponding to time t=nh=7.8. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
y′=−ay; yₙ=(1−ah)ⁿ; exact y(nh)=e^(−anh)
03 · Reflect and transfer
Explain what changes and why.
A stable Euler calculation can still be inaccurate. What happens to accuracy when the step decreases while final physical time is held fixed?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.