G: Differentiation: Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

The idea

Constructing a differential equation means translating a written or physical description into an equation involving a derivative. Identify the changing quantities and the independent variable, then express the given rate of change as a derivative. For example, if a population P grows at a rate proportional to its current size, write dP/dt = kP. In kinematics, acceleration is dv/dt or d²x/dt². For price-demand, if demand falls at a rate proportional to the excess of price over a threshold, write dD/dt = -k(P - P0). Always define symbols and include the constant of proportionality. The equation should capture the relationship, not solve it. In pure mathematics, you may be given a family of curves and asked to form the differential equation by eliminating parameters. The skill is assessed by checking that the derivative is correctly identified and the relationship is accurately represented.

What earns the marks

• Correctly identify the dependent and independent variables from the context.
• Express the rate of change as a derivative (e.g., dP/dt, dv/dt, dD/dp).
• Include a constant of proportionality where the relationship is proportional.
• Use correct signs to reflect increase or decrease (e.g., negative for decay).
• For pure mathematics, eliminate arbitrary constants to form the differential equation.

Common mistakes and how to put them right

• Using the wrong derivative.

  Put it right: For example, writing dP/dx instead of dP/dt when time is the independent variable. Always check which variable is changing with respect to which.

• Forgetting the constant of proportionality.

  Put it right: If a rate is 'proportional to', a constant k must be included. Omitting it loses the generality of the relationship.

• Incorrect sign: if a quantity decreases, the derivative should be negative.

  Put it right: For example, demand decreasing as price increases gives dD/dp = -k, not +k.

Exam tips

• Underline key phrases such as 'rate of change', 'proportional to', and 'with respect to' to identify the derivative and variables.
• Define your symbols clearly at the start of your answer, especially in context questions, to avoid ambiguity.
• Check that your equation makes sense dimensionally and that the signs match the described behaviour.