Constructing differential equations from rate laws
| English | 中文 | Pinyin |
|---|---|---|
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | 微分方程 | wēi fēn fāng chéng |
A population grows in proportion to its current size while a fixed number leaves each day. Which parts belong in the rate equation?
- A population grows in proportion to its current size while a fixed number leaves each day. Which parts belong in the rate equation?
- This lesson studies differential equation 微分方程: A relationship involving an unknown function and one or more of its derivatives.
Choose the mathematical structure
- Choose independent and dependent variables and convert the stated rate law into a derivative. Proportional to y means ky; proportional to y² means ky². A fixed inflow minus proportional loss gives dy/dt=a−ky; a rate proportional to the gap from a target A gives dy/dt=k(A−y). An initial condition fixes a function value, not a derivative formula. Constants must have units making both sides the same kind of rate.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines differential equation?
A relationship involving an unknown function and one or more of its derivatives.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For an illustrative population P, t in days, births occur at 0.04P per day and 10 individuals leave per day. The model is dP/dt=0.04P−10, P(0)=500. Initially the rate is 10 individuals/day; P=250 is an equilibrium, and below it the same equation predicts decline. The model must stop or be refined if it predicts a negative population. For a velocity v in m/s with t in seconds, acceleration 6−0.2v gives dv/dt=6−0.2v, v(0)=0: initial acceleration is 6 m/s² and equilibrium speed 30 m/s. In a hypothetical demand model, quantity Q falls at a price-rate proportional to Q. At price p=12, Q=200 and dQ/dp=−10, so dQ/dp=−0.05Q with Q(12)=200; the independent variable is price, not time. In pure maths, slope 2xy with y(1)=3 gives dy/dx=2xy and initial slope 6.
Constructing differential equations from rate laws
Choose independent and dependent variables and convert the stated rate law into a derivative
Connect derivative calculations to their original point, domain and stated rate law.
Find the initial population rate in individuals/day.
0.04×500−10=10.
Test a tempting shortcut
- Proportional rate is not constant absolute change. Fixed departures need a subtraction term, not a second multiplier of P. Do not replace a changing P with its initial value throughout the equation. An equilibrium solves the rate-equals-zero condition; it need not equal the initial value or be approached from every start. A price derivative and a time derivative are different quantities with different units.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A statement that the rate is proportional to the current amount means the same absolute change at all times. This claim is false. Explain which definition or assumption it violates.
Find the initial acceleration in m/s².
6−0.2×0=6.
A statement that the rate is proportional to the current amount means the same absolute change at all times.
Proportional rate is not constant absolute change. Fixed departures need a subtraction term, not a second multiplier of P. Do not replace a changing P with its initial value throughout the equation. An equilibrium solves the rate-equals-zero condition; it need not equal the initial value or be approached from every start. A price derivative and a time derivative are different quantities with different units.
Interpret a new situation
- Define each variable, translate every inflow/outflow or proportional term with its sign, state constant assumptions and attach the initial condition separately. Evaluate the initial rate as a check and find any equilibrium before interpreting the model. These are illustrative constructions, not compulsory context lists: G6 examples are optional. Solving the resulting equation is a separate integration task.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the positive proportional constant in the demand model.
From −10=−k×200, k=10/200=0.05.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · G. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A relationship involving an unknown function and one or more of its derivatives. Choose the relationship, show the method, check its assumptions and interpret the result.