A2 June 2023 Paper 1 Q15
15 Find the general solution of the differential equation
\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 3\frac{\mathrm{d}y}{\mathrm{d}x} - 4y = \cos 2x + 5x\][9 marks]
| Scheme | Marks | AO |
|---|---|---|
| Uses auxiliary equation | M1 | 3.1a |
| Obtains correct complementary function | A1 | 1.1b |
| Uses correct form of particular integral | M1 | 3.1a |
| Obtains first and second derivatives of their particular integral | M1 | 1.1a |
| Substitutes first and second derivatives of particular integral into DE | M1 | 3.1a |
| Forms a pair of simultaneous equations by comparing coefficients of \(\cos 2x\) and \(\sin 2x\) and solves them | M1 | 3.1a |
| Forms a pair of simultaneous equations by comparing coefficient of \(x\) and constant term and solves them | M1 | 3.1a |
| Obtains correct trig constants or linear constants | A1 | 1.1b |
| Completes a reasoned argument to obtain completely correct solution, including \(y =\) | R1 | 2.1 |
| (9 marks) |