FP2 June 2008 Q1
1. Solve the differential equation \[\frac{\mathrm{d}y}{\mathrm{d}x} - 3y = x\] to obtain \(y\) as a function of \(x\). (5)
| Scheme | Marks |
|---|---|
| Integrating factor \(= \mathrm{e}^{-3x}\) | B1 |
| \(\therefore \dfrac{\mathrm{d}}{\mathrm{d}x}(y\mathrm{e}^{-3x}) = x\mathrm{e}^{-3x}\) | M1 |
| \(\therefore (y\mathrm{e}^{-3x}) = \displaystyle\int x\mathrm{e}^{-3x}\,\mathrm{d}x = -\dfrac{x}{3}\mathrm{e}^{-3x} + \int \dfrac{1}{3}\mathrm{e}^{-3x}\,\mathrm{d}x\) | M1 |
| \(= -\dfrac{x}{3}\mathrm{e}^{-3x} - \dfrac{1}{9}\mathrm{e}^{-3x}\ (+c)\) | A1 |
| \(\therefore y = -\dfrac{x}{3} - \dfrac{1}{9} + c\mathrm{e}^{3x}\) | A1ft |
| (5) | |
| (5 marks) |
Notes
First M for multiplying through by Integrating Factor and evidence of calculus
Second M for integrating by parts ‘the right way around’.
Be generous – ignore wrong signs and wrong constants.
Second M dependent on first. Both As dependent on this M.
First A1 for correct expression – constant not required
Second A requires constant for follow through.
If treated as a second order de with errors then send to review.