Constructing differential equations from rate laws
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/ | phương trình vi phân |
Which parts belong in the rate equation?
- A population grows in proportion to its current size while a fixed number leaves each day.
- This lesson studies differential equation 微分方程: A relationship involving an unknown function and one or more of its derivatives.
The method
- 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.
Why is "rate proportional to the current amount" written dy/dt=ky rather than dy/dt=k?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A proportional rate gives exponential behaviour, not constant absolute change. dy/dt=k would add a fixed amount per unit time regardless of y - a linear model answering a different statement.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
0.04×500−10=10.
Where it goes wrong
- 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.
Not true: A statement that the rate is proportional to the current amount means the same absolute change at all times. Proportional to the current amount means dN/dt=kN, so the change is larger when N is larger.
Find the initial acceleration in m/s².
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
6−0.2×0=6.
A statement that the rate is proportional to the current amount means the same absolute change at all times.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Proportional to the current amount means dN/dt=kN, so the change is larger when N is larger.
Applying it
- 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.
Find the positive proportional constant in the demand model.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
From −10=−k×200, k=10/200=0.05.
Match each rate law to its differential equation.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Constants carry units that make both sides the same kind of rate; an initial condition fixes a VALUE, never the derivative formula.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Differential equation: A relationship involving an unknown function and one or more of its derivatives.
- Choose independent and dependent variables and convert the stated rate law into a derivative.
- Watch for: proportional rate is not constant absolute change.