| Term | Meaning / Usage |
|---|---|
| Differential Equation | An equation involving derivatives that describes how one quantity changes with respect to another. |
| General Solution | A family of solutions containing an arbitrary constant that represents infinitely many possible curves. |
| Separable Equation | A differential equation that can be rearranged so all y-terms and x-terms are separated. |
📌 Setting Up Differential Equations from Context
- Verbal phrases such as “rate of change” must always be translated into derivative notation like dG/dt.
- Statements involving proportionality require introducing a constant, forming equations such as dG/dt = kG.
- Descriptions where growth slows as quantity increases often imply square-root, quadratic, or logistic relationships.
- All variables must be clearly defined with units before forming equations to ensure physical and mathematical validity.
🌍 Real-World Connection
- Population growth is often modelled using proportional relationships where growth rate depends directly on population size.
- Radioactive decay processes follow differential equations where decay rate depends on remaining quantity.
- Chemical reaction rates are commonly modelled using concentration-based differential equations.
📌 Solving by Separation of Variables
- The equation must first be rearranged so all y-terms multiply dy and all x-terms multiply dx.
- Both sides are then integrated independently, producing an equation relating integrated expressions.
- A single constant of integration is introduced after completing both integrations.
- The final step involves rearranging the equation to express y explicitly when possible.
📌 Interpreting Contextual Differential Equation Questions
- Keywords such as “grows”, “decays”, or “proportional” directly indicate derivative-based relationships.
- The dependent variable usually represents the quantity being modelled, while time is independent.
- The sign of the proportionality constant determines whether growth or decay is occurring.
- Checking units ensures the mathematical model is consistent with the real-world situation.
📐 IA Spotlight
- Students can model biological growth, cooling processes, or chemical decay using separable equations.
- Strong investigations justify why proportional or nonlinear models best represent observed behaviour.
- Evaluation should include limitations of assumptions, particularly long-term proportionality.
🌐 EE Focus
- Extended Essays may compare exponential and logistic models using real experimental data.
- Parameter sensitivity analysis can assess how constants influence long-term predictions.
- Research may evaluate model accuracy against real biological or environmental systems.
📌 Example 1 — Setting Up a Model
Context: An algae population G grows at a rate proportional to √G.
- The phrase “rate proportional to √G” translates directly to dG/dt = k√G.
- Separating variables places G-terms on one side and time-terms on the other.
- Integrating both sides produces a general solution involving an arbitrary constant.
Result: √G = (kt + C)/2
📌 Example 2 — Exponential Differential Equation
- Exponential growth occurs when the rate of change depends directly on current size.
- Separation yields an equation involving the natural logarithm.
- Initial conditions are applied after integration to determine constants.
Final solution: y = 5ekx
📝 Paper Tip
- All modelling questions must begin with clearly stated variable definitions.
- Method marks are awarded for clear separation steps, not only final answers.
- Constants of integration must always appear unless a particular solution is required.
📌 Multiple Choice Questions (MCQs)
MCQ 1
Which differential equation correctly represents a quantity y that increases at a rate proportional to its current value?
- A. dy/dx = y + k
- B. dy/dx = ky
- C. dy/dx = kx
- D. dy/dx = y/k
Answer & Explanation
Correct answer: B
The phrase “rate proportional to its current value” translates directly to dy/dx ∝ y.
Introducing a constant of proportionality k gives dy/dx = ky.
This is the standard exponential growth or decay model used in population growth, finance, and radioactive decay.
MCQ 2
Which of the following differential equations is separable?
- A. dy/dx = x + y
- B. dy/dx = xy
- C. dy/dx + y = x
- D. dy/dx = x/y + y
Answer & Explanation
Correct answer: B
The equation dy/dx = xy can be rearranged as (1/y)dy = x dx, which separates all y-terms on one side and x-terms on the other.
The other equations cannot be rearranged into this form without additional techniques.
MCQ 3
When solving a separable differential equation, when should the constant of integration be introduced?
- A. Before separating variables
- B. After integrating both sides
- C. After solving for y
- D. Before applying initial conditions
Answer & Explanation
Correct answer: B
The constant of integration must be added immediately after integrating both sides of the equation.
Introducing it earlier or later risks losing generality or making algebraic errors.
📌 Short Answer Questions
Short Question 1
Explain what is meant by a general solution of a differential equation.
Model Answer
A general solution is a family of functions that satisfies a given differential equation and contains an arbitrary constant.
This constant represents infinitely many possible curves, each corresponding to a different initial condition.
A specific solution is obtained only when sufficient initial conditions are provided to determine the constant.
Short Question 2
State two features that indicate a differential equation can be solved by separation of variables.
Model Answer
First, the equation must allow rearrangement so that all y-terms multiply dy and all x-terms multiply dx.
Second, no term should involve both x and y added together, as this prevents clean separation.
Equations of the form dy/dx = f(x)g(y) are always separable.
📌 Long Answer Questions (IB AIHL Style)
Long Question 1
A population of bacteria P grows such that the rate of increase of the population is proportional to the size of the population at time t.
(a) Form a differential equation to model this situation, clearly defining all variables and constants.
(b) Solve the differential equation to obtain the general solution.
(c) Given that P = 200 when t = 0, find the particular solution.
(d) Explain what the constant of proportionality represents in the context of the problem.
Full Worked Solution
(a) Forming the model
Let P represent the population size and t represent time.
“Rate of increase proportional to population” implies dP/dt ∝ P.
Introducing a constant k gives the differential equation dP/dt = kP.
(b) Solving the differential equation
Separate variables: (1/P)dP = k dt.
Integrate both sides to obtain ln|P| = kt + C.
Exponentiating gives P = Aekt.
(c) Applying the initial condition
Using P(0) = 200 gives A = 200.
Hence, P = 200ekt.
(d) Interpretation
The constant k represents the growth rate per unit time.
A larger k indicates faster population growth, while a negative value would indicate decay.
Long Question 2
The temperature T of a cooling object decreases at a rate proportional to its temperature at time t.
(a) Write a differential equation representing this situation.
(b) Solve the differential equation to obtain the general solution.
(c) If T = 80 when t = 0 and T = 40 when t = 10, find the value of the proportionality constant.
(d) Comment on the physical meaning of the negative sign in the model.
Full Worked Solution
The negative sign indicates cooling, meaning temperature decreases over time as heat is lost to the surroundings.
This aligns with real physical cooling processes described by Newton’s Law of Cooling.