FP2 January 2006 Q7
7. \[(1 + 2x)\frac{\mathrm{d}y}{\mathrm{d}x} = x + 4y^2.\]
(a) Show that \[(1 + 2x)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 1 + 2(4y - 1)\frac{\mathrm{d}y}{\mathrm{d}x} \qquad \boxed{1}\] (2)
(b) Differentiate equation \(\boxed{1}\) with respect to \(x\) to obtain an equation involving \[\frac{\mathrm{d}^3y}{\mathrm{d}x^3},\ \frac{\mathrm{d}^2y}{\mathrm{d}x^2},\ \frac{\mathrm{d}y}{\mathrm{d}x},\ x \text{ and } y.\] (3)
Given that \(y = \tfrac{1}{2}\) at \(x = 0\),
(c) find a series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x^3\). (6)
| Scheme | Marks |
|---|---|
| Correct method for producing 2nd order differential equation e.g. \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left\{(1 + 2x)\dfrac{\mathrm{d}y}{\mathrm{d}x}\right\} = \dfrac{\mathrm{d}}{\mathrm{d}x}\left\{x + 4y^2\right\}\) attempted | M1 |
| \((1 + 2x)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2\dfrac{\mathrm{d}y}{\mathrm{d}x} = 1 + 8y\dfrac{\mathrm{d}y}{\mathrm{d}x}\) seen + conclusion AG | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Differentiating again w.r.t. \(x\): \((1 + 2x)\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3} + 2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 8y\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 8\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2 - 2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) or equiv. [e.g. \((1 + 2x)\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3} = 8\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 4(2y - 1)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\)] | M1 A2, 1, 0 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) (at \(x = 0\)) \(= 1\) | B1 |
| Finding \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) (at \(x = 0\)) \((= 3)\) | M1 |
| Finding \(\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}\), at \(x = 0\); \(= 8\) [A1 f.t. is on part (c) values only] | M1 A1ft |
| \(y = \dfrac{1}{2} + x + \dfrac{3}{2}x^2 + \dfrac{4}{3}x^3 + \ldots\) | M1 A1 |
| (6) | |
| (11 marks) |
Notes
[Alternative (c):
| Scheme | Marks |
|---|---|
| Polynomial for \(y\): \(y = \tfrac{1}{2} + ax + bx^2 + cx^3 + \ldots\) | M1 |
| In given d.e.: \((1 + 2x)(a + 2bx + 3cx^2 + \ldots) \equiv x + 4(\tfrac{1}{2} + ax + bx^2 + cx^3 + \ldots)^2\) | M1A1 |
| \(a = 1\) B1, Complete method for other coefficients M1, answer A1] |