FP2 June 2008 Q3
3.
(a) Find the general solution of the differential equation \[3\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - \frac{\mathrm{d}y}{\mathrm{d}x} - 2y = x^2\] (8)
(b) Find the particular solution for which, at \(x = 0\), \(y = 2\) and \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\). (6)
| Scheme | Marks |
|---|---|
| Solve auxiliary equation \(3m^2 - m - 2 = 0\) to obtain \(m = -\dfrac{2}{3}\) or 1 | M1A1 |
| C.F is \(A\mathrm{e}^{-\frac{2}{3}x} + B\mathrm{e}^{x}\) | A1ft |
| Let PI \(= \lambda x^2 + \mu x + \nu\). Find \(y\prime = 2\lambda x + \mu\), and \(y\prime\prime = 2\lambda\) and substitute into d.e. | M1 |
| Giving \(\lambda = -\dfrac{1}{2}\), \(\mu = \dfrac{1}{2}\) and \(\nu = -\dfrac{7}{4}\), | A1A1A1 |
| \(\therefore y = -\dfrac{1}{2}x^2 + \dfrac{1}{2}x - \dfrac{7}{4} + A\mathrm{e}^{-\frac{2}{3}x} + B\mathrm{e}^{x}\) | A1ft |
| (8) |
Notes
Attempt to solve quadratic expression with 3 terms (usual rules)
Both values required for first accuracy.
Real values only for follow through
Second M 3 term quadratic for PI required
Final A1ft for their CF+ their PI dependent upon at least one M
| Scheme | Marks |
|---|---|
| Use boundary conditions: | |
| \(2 = -\dfrac{7}{4} + A + B\) | M1A1ft |
| \(y\prime = -x + \dfrac{1}{2} - \dfrac{2}{3}A\mathrm{e}^{-\frac{2}{3}x} + B\mathrm{e}^{x}\) and \(3 = \dfrac{1}{2} - \dfrac{2}{3}A + B\) | M1A1 |
| Solve to give \(A = 3/4\), \(B = 3\) \(\left(\therefore y = -\dfrac{1}{2}x^2 + \dfrac{1}{2}x - \dfrac{7}{4} + \dfrac{3}{4}\mathrm{e}^{-\frac{2}{3}x} + 3\mathrm{e}^{x}\right)\) | M1 A1 |
| (6) | |
| (14 marks) |
Notes
Second M for attempt to differentiate their \(y\) and third M for substitution