Differential equations in A Level Math Paper 3: common mistakes and examiner advice

Differential equation questions in Paper 3 follow a fixed pattern: form or separate the equation, integrate both sides, find the constant, and then use the solution. The reports show that students know the pattern. The marks are lost in the integration details, in the constant, and in the final rearrangement. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Sign and coefficient errors in the integration. This is the most repeated comment. Typical examples: omitting a minus sign from one or both integrals, getting ln t when the answer should have been −½ ln 2t, putting a 3 or a 2 in the wrong place (for example the 2 from the denominator of y⁻² ending up in the numerator), and a wrong coefficient for a trigonometric term.
• Integrating by the wrong rule. Writing ∫e⁻ʸ dy = ln(eʸ), e⁻ˣ integrated as e⁻ˣ, tan x integrated as sec x, a sign error when integrating 2 tan x, and not writing 1/cos²y as sec²y so that it integrates to tan y. Trying a variety of approaches to ∫(x + 2)/(x + 1) dx, most of them incorrect, instead of dividing or rewriting the numerator.
• Separating the variables incorrectly. After separating, dx and dy were not always in the numerator, or a function was inverted. Some students moved the constant k to the other side before integrating and made errors in the coefficient.
• The constant of integration. Many things go wrong: it is omitted (for example the constant 10 when the integration produced a ln u and a u term), it is confused with k, it is implied to equal 1, or an incorrect value is used. In a context question, students used values other than the given starting values (such as t = 0 and V = 0) to find the constant.
• The final rearrangement. Using e^(a+b) = eᵃ + eᵇ is incorrect. Reports also mention errors in moving e⁻¹ and 1/e within the equation obtained from a second boundary condition, and an incorrect procedure when removing logarithms. An answer should not be left in the form e^ln….
• Not finishing. Students did not state the solution in terms of the two variables asked for (such as V and t), did not substitute the value of x or t given in the question, or did not give the final statement, such as y = 2 ln(…).
• Not using the earlier part. In some multi-part questions, students did not realise that they needed to solve the differential equation stated in an earlier part.
• Exact answers. Where the question asks for an exact value, a decimal answer does not gain full marks.

What examiners advise

• Write dx and dy in the numerator and check the integrals on each side separately.
• Include the constant straight away and find it from the given condition, not from another value.
• Use the formula book for the trigonometric identities and integrals that you cannot recall. One report notes that incorrect signs in a trigonometric formula could have been avoided this way.
• Check that your final answer is in the form asked for, such as y in terms of x, or V in terms of t.
• If you are working towards a given answer and do not reach it, go back to find the error.

Habits that help

• Differentiate your final integral to see if it gives back the integrand.
• Say what each letter stands for before you begin: constant, rate, variable.
• Substitute the initial condition into your final equation as a check.

Questions students ask

What are the most common mistakes in differential equation questions?

Sign and coefficient errors in the integration. Examiners call it the most repeated comment: a missing minus sign, a 2 or a 3 in the wrong place, or integrating tan x as sec x or e⁻ˣ as e⁻ˣ.

How do I separate the variables correctly?

Get dx and dy into the numerators and check the integral on each side separately. Reports record students inverting a function, or moving the constant k to the other side before integrating and then making coefficient errors.

When do I add the constant of integration?

Straight away, and find it from the given condition, not from another value. Reports list the constant being omitted, confused with k, assumed to be 1 or given an incorrect value.

Is e^(a + b) equal to eᵃ + eᵇ?

No. It is eᵃ × eᵇ. Using e^(a+b) = eᵃ + eᵇ is a recorded error in the final rearrangement.

How should I finish a differential equation question?

State the solution in terms of the variables asked for, such as V and t, substitute the value given in the question, and give an exact answer when one is asked for. Check the answer is in the form requested, such as y in terms of x.