Undergraduate · Euler stability · 527 of 650
Euler stability · Decay rate a=1.75; Step size h=1.5
Use n=6 updates, corresponding to time t=nh=9. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.75; Step size h=1.5.
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′=−1.75y, y₀=1, with fixed step h=1.5. Use n=6 updates, corresponding to time t=nh=9. Find the Euler value, exact value at the same time, and stability classification from the amplification factor.
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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′=−1.75y, y₀=1, with fixed step h=1.5. Use n=6 updates, corresponding to time t=nh=9. 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−(1.75)(1.5)=-1.625. After n=6 steps, yₙ=(-1.625)^6=18.412815; the exact solution at time nh=9 is exp(−1.75(9))=0.
Step 3
Check the mathematical meaning
The signed factor is -1.625. Its magnitude 1.625 is above one, so repeated numerical magnitudes grow. The exact solution always decays for a>0. Euler step: 6; Computed value: 18.413; Exact value: 0; Amplification |1−ah|: 1.625: unstable growth. Decimal values are rounded, so use unrounded intermediate values.
The answer
One Euler update multiplies by 1−(1.75)(1.5)=-1.625. After n=6 steps, yₙ=(-1.625)^6=18.412815; the exact solution at time nh=9 is exp(−1.75(9))=0. The signed factor is -1.625. Its magnitude 1.625 is above one, so repeated numerical magnitudes grow. The exact solution always decays for a>0. Animation check: Euler step: 6; Computed value: 18.413; Exact value: 0; Amplification |1−ah|: 1.625: unstable growth. 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.
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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′=−1.75y, y₀=1, with fixed step h=1.5. Use n=6 updates, corresponding to time t=nh=9. 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.