Differential equations: a law of change as a process model

Where we are on the map

The starting information is now a relationship between a quantity and its rate of change.

Begin with a familiar image

When an account repeatedly earns a percentage of its current balance, the addition grows with the amount already present. “Rate is proportional to amount” becomes y′=ky.

Precise meaning

A differential equation relates an unknown function to its derivatives. The family y=Ce^{kt} solves y′=ky; the initial condition y(0)=y₀ selects C=y₀. The sign of k determines growth or decay, and its unit is inverse time.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

quantities → verbal law → equation → solution family → initial condition → check → model limits

Worked example

For y′=0.05y, y(0)=100, the solution is y=100e^{0.05t}. Check: y′=5e^{0.05t}=0.05y, and the value at t=0 is 100.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Using a positive k for decay and never checking behavior is wrong. Substitution, sign, units, and the long-term limit must agree with the process.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. A substance follows M′=−0.3M, M(0)=50. Find M(t), the time to reach 10 units, and check by differentiation and monotonicity.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The checkpoint combines limits, derivatives, optimization, integrals, and differential models in one system.

Course contents