Differential Equations: Exam Structures
Introduction: How to Approach Differential Equation Modelling Questions
Many real-life situations can be modelled using differential equations. In H2 Mathematics, these questions commonly appear in contexts such as population growth, cooling, finance, medicine, radioactive decay, and motion.
Before rushing into calculations, students should follow a structured thinking process.
Step 1 — Form the Differential Equation
After reading the entire question, identify:
- What quantity is changing
- What the rate depends on
- Whether there are increasing and decreasing effects
A common modelling structure is:
dx/dt = (rate of increase) − (rate of decrease) For example:
- Population growth with deaths
- Water flowing into and out of a tank
- Medicine entering and leaving the bloodstream
Unknown constants such as k usually appear in the equation.
Step 2 — Solve the Differential Equation
Most examination questions involve:
- Variable separable differential equations
- Integration
- Using logarithms
Always remember to:
- Separate variables carefully
- Integrate both sides
- Include the integrating constant
c - Simplify fully before proceeding
Step 3 — Use Initial Conditions
Words such as:
- “Initially”
- “When t = 0”
- “At the start”
usually provide information to solve for constants like:
c and k
Step 4 — Answer the Final Context Question
Examiners often ask students to:
- Find long-term behaviour
- Sketch graphs
- Determine maximum/minimum values
- Interpret physical meaning
- Find time taken to reach a value
Always check whether your answer makes sense in context.
For example:
- Population cannot become negative
- Mass cannot exceed physical limits
- Time values may need rejection
Free Differential Equations Lesson
For a full free lesson on Differential Equations, visit:
https://www.musclemathtuition.com/differential-equations-lesson-2/
Common Exam Structure Example
Question
A tank initially contains 100 litres of water.
Water enters the tank at a constant rate of 8 litres per minute.
Water leaves the tank at a rate proportional to the amount of water currently inside the tank.
If x litres of water are in the tank after t minutes, and the constant of proportionality is 0.04, form and solve the differential equation.
Find the amount of water in the tank after 20 minutes.
Step-by-Step Solution
Step 1 — Form the Differential Equation
We identify:
- Rate of increase = 8
- Rate of decrease = 0.04x
Hence:
dx/dt = 8 − 0.04x This is one of the most common H2 Math modelling structures.
Step 2 — Separate Variables
Rearrange:
dx / (8 − 0.04x) = dt Integrate both sides:
∫ dx / (8 − 0.04x) = ∫ dt
Step 3 — Integrate
After integration:
−25 ln|8 − 0.04x| = t + c Rearranging:
ln|8 − 0.04x| = −0.04t + c Exponentiating:
8 − 0.04x = Ae−0.04t
Step 4 — Use Initial Condition
Initially:
x = 100 when t = 0 Substitute:
8 − 0.04(100) = A A = 4 Hence:
8 − 0.04x = 4e−0.04t Rearranging:
x = 200 − 100e−0.04t
Step 5 — Find the Amount After 20 Minutes
Substitute t = 20:
x = 200 − 100e−0.8 Using calculator evaluation:
x ≈ 155.1 litres
Common Differential Equations Exam Structures Students Must Know
1. Growth and Decay Models
Usually appears in:
- Population questions
- Radioactive decay
- Bacteria growth
- Compound interest modelling
Typical structure:
dx/dt = kx or
dx/dt = −kx
2. Limiting Value Models
These involve a quantity approaching a maximum or equilibrium value.
Typical structure:
dx/dt = a − bx Very common in:
- Temperature questions
- Water tanks
- Drug concentration models
3. Logistic-Type Models
Used when growth slows near a limiting capacity.
Typical structure:
dx/dt = kx(M − x) Students are often asked to:
- Find equilibrium values
- Sketch solution curves
- Interpret long-term behaviour
4. Context-Based Modelling Questions
These are the most difficult because students must first create the equation themselves.
Common phrases include:
- “Rate proportional to…”
- “Varies directly with…”
- “Net rate of change…”
- “Rate decreases as…”
The biggest mistake students make is jumping into integration before understanding the model structure.
Final Advice for H2 Mathematics Students
Differential Equations questions are highly structured once students recognise the recurring patterns.
The strongest students are usually not the fastest integrators — they are the students who can:
- Identify the correct modelling structure
- Interpret wording carefully
- Apply initial conditions accurately
- Understand long-term behaviour
If you struggle with Differential Equations, focus first on recognising the common exam structures before memorising procedures.
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