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AQA A-Level Mathematics: Separation of Variables for First Order Differential Equations — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 2 July 2026

The short answer

When solving first-order differential equations, one common method is separation of variables. This technique involves rearranging the equation so that all terms involving the dependent variable (usually y ) are on one side and all terms involving the independent variable (usually x ) are on the other side.

The question

Solve the differential equation dy/dx = 3y/x. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

S1
1. Identify the equation: dy/dx = 3y/x.
S2
2. Separate the variables: (1/y) dy = (3/x) dx.
S3
3. Integrate both sides: ∫(1/y) dy = ∫(3/x) dx.
S4
4. The result of integration is ln|y| = 3ln|x| + C, which can be written as ln|y| = ln|x 3 | + C.
S5
5. Exponentiate both sides: |y| = e ln|x 3 | + C = A|x 3 |, where A = ±e C .
S6
6. The general solution is y = Ax 3 .

Model answer

Worked through, with each step tagged to the mark it earns.

S1
1. Identify the equation: dy/dx = 3y/x.
S2
2. Separate the variables: (1/y) dy = (3/x) dx.
S3
3. Integrate both sides: ∫(1/y) dy = ∫(3/x) dx.
S4
4. The result of integration is ln|y| = 3ln|x| + C, which can be written as ln|y| = ln|x 3 | + C.
S5
5. Exponentiate both sides: |y| = e ln|x 3 | + C = A|x 3 |, where A = ±e C .
S6
6. The general solution is y = Ax 3 .
Final answer: y = Ax 3

Common mistakes

Forgetting to separate the variables before integrating. — Always ensure that all terms involving y are on one side and all terms involving x are on the other before integrating.
Forgetting to include the constant of integration C. — Always include a constant of integration C when integrating both sides of the equation.
Incorrectly solving for y after exponentiating both sides. — Double-check your algebra when solving for y. Ensure that you correctly handle the absolute value and constants.
Applying initial conditions incorrectly to find the particular solution. — Substitute the initial values carefully and solve for the constant C. Double-check your calculations.
Forgetting to factorise when separating variables. — Factorise any common factors before separating the variables. This ensures that all terms are correctly separated.
Incorrectly handling absolute values when integrating 1/y. — Always use ln|y| when integrating (1/y) dy. This ensures that the solution is valid for all y ≠ 0.
Forgetting to check the domain of the solution. — Always check the domain of the solution to ensure it is valid for the given problem. Consider any restrictions on x and y.

Where the marks go

Full worked solution (all marking points)5 marks