The same kind of equation describes a rabbit population exploding, a radioactive sample fading, and a spring bouncing back and forth. Solving them by separating variables reveals the shared math behind growth, decay, and oscillation.
Solving the simple ones
For y' = ky, separate the variables — put all the y's on one side and x's on the other — then integrate to get y = Ce^{kx}. Positive k gives growth (populations, compound interest); negative k gives decay (cooling coffee, radioactive atoms). Second-order equations like y'' = −y produce oscillation: springs and waves.
y = 1x² · slope at x=1 (the derivative 2a) = 2
Watch how a curve's slope (its derivative) changes point to point — the raw material every differential equation works with.
- Write Newton's cooling law as y' = −k(y − room), where y is the coffee's temperature.
- Explain why the coffee cools fast at first, then slower.
- Predict the temperature the coffee approaches after a long time.
- Say whether k is positive or negative here, and why.
What you should see: You connected a first-order ODE to a real cooling curve and read its long-term behavior.