Differential Equations: The Math of Cooling Coffee and Swaying Bridges
How equations about rates of change describe growth, decay, oscillation, resonance and heat, and the surprises hidden in their solutions.
Differential Equations · 50 questions
Try 3 questions
1. The equation dy/dt = ky with k > 0 describes what kind of behavior?
- Steady linear growth
- Growth that levels off
- Exponential growth
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Answer: Exponential growth
The rate of increase is proportional to the current amount, so the solution is y = y₀e^(kt), doubling at fixed intervals.
2. A radioactive sample has a half-life of 10 years. What fraction of it remains after 30 years?
- None of it
- One quarter
- One eighth
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Answer: One eighth
Exponential decay halves the amount every 10 years: 1/2, then 1/4, then 1/8 after three half-lives. It never quite reaches zero.
3. What makes an ordinary differential equation 'linear'?
- Its solution graph is a straight line
- No powers or products of the unknown
- It contains only first derivatives
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Answer: No powers or products of the unknown
Linear means no y², y·y′, sin y and so on; coefficients may still depend on x, and the solutions need not be straight lines.
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