FP2 June 2005 Q3
3.
(a) Show that the transformation \(y = xv\) transforms the equation \[x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2x\frac{\mathrm{d}y}{\mathrm{d}x} + (2 + 9x^2)y = x^5, \qquad \text{I}\] into the equation \[\frac{\mathrm{d}^2v}{\mathrm{d}x^2} + 9v = x^2. \qquad \text{II}\] (5)
(b) Solve the differential equation II to find \(v\) as a function of \(x\). (6)
(c) Hence state the general solution of the differential equation I. (1)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = x\dfrac{\mathrm{d}v}{\mathrm{d}x} + v,\ \ \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = x\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 2\dfrac{\mathrm{d}v}{\mathrm{d}x}\) [M1 for diff. product, A1 both correct] | M1A1 |
| \(\therefore x^2\left(x\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 2\dfrac{\mathrm{d}v}{\mathrm{d}x}\right) - 2x\left(x\dfrac{\mathrm{d}v}{\mathrm{d}x} + v\right) + (2 + 9x^2)vx = x^5\) | M1 |
| \(x^3\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 2x^2\dfrac{\mathrm{d}v}{\mathrm{d}x} - 2x^2\dfrac{\mathrm{d}v}{\mathrm{d}x} - 2vx + 2vx + 9vx^3 = x^5\) \(\left[x^3\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + {} + 9vx^3 = x^5\right]\) | A1 |
| Given result: \(\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 9v = x^2\) cso | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| CF: \(v = A\sin 3x + b\cos 3x\) (may just write it down) | M1A1 |
| Appropriate form for PI: \(v = \lambda x^2 + \mu\) (or \(ax^2 + bx + c\)) | M1 |
| Complete method to find \(\lambda\) and \(\mu\) (or \(a, b, c\)) | M1 |
| \(v = A\sin 3x + B\cos 3x + \dfrac{1}{9}x^2 - \dfrac{2}{81}\) [f.t. only on wrong CF] | M1A1ft |
| (6) |
| Scheme | Marks |
|---|---|
| \(\therefore y = Ax\sin 3x + Bx\cos 3x + \dfrac{1}{9}x^3 - \dfrac{2}{81}x\) [f.t. for \(y = x\) (candidate’s CF + PI), providing two arbitrary constants] | B1ft |
| (1) | |
| (12 marks) |