FP2 June 2007 Q1
1. Obtain the general solution of the differential equation \[x\frac{\mathrm{d}y}{\mathrm{d}x} + 2y = \cos x, \qquad x \gt 0,\] giving your answer in the form \(y = \mathrm{f}(x)\).
| Scheme | Marks |
|---|---|
| Attempt to arrange in correct form \(\dfrac{\mathrm{d}y}{\mathrm{d}x} + \dfrac{2}{x}y = \dfrac{\cos x}{x}\) | M1 |
| Integrating Factor: \(= \mathrm{e}^{\int \frac{2}{x}\mathrm{d}x},\ \left[(= \mathrm{e}^{2\ln x} = \mathrm{e}^{\ln x^2}) = x^2\right]\) \(\left[x^2\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2xy = x\cos x\right.\) implies M1M1A1] | M1, A1 |
| \(\therefore x^2y = \displaystyle\int x^2 \cdot \frac{\cos x}{x}\,\mathrm{d}x\) or equiv. [IF. \(y = \int I.F.\) (candidate’s RHS)\(\mathrm{d}x\)] | M1ft |
| By Parts: \((x^2y) = x\sin x - \int \sin x\,\mathrm{d}x\) | M1 |
| i.e. \((x^2y) = x\sin x, + \cos x\ (+c)\) | A1, A1cao |
| \(y = \dfrac{\sin x}{x} + \dfrac{\cos x}{x^2} + \dfrac{c}{x^2}\) | A1ft |
| (8 marks) |
Notes
First M: At least two terms divided by \(x\).
“By parts” M: Must be complete method, e.g \(\int x^2\cos x\,\mathrm{d}x\) requires two applications
Because of functions involved, be generous with sign, but \(x\sin x \pm \int \cos x\,\mathrm{d}x\) is M0
(S.C. “Loop” integral like \(\int \mathrm{e}^x\cos x\,\mathrm{d}x\), allow M1 if two applications of “by parts”, despite incomplete method)
Final A ft for dividing all terms by candidates IF., providing “c” used.