C4 January 2010 Q5
5.
(a) Find \(\displaystyle\int \frac{9x + 6}{x}\,\mathrm{d}x,\ x > 0\). (2)
(b) Given that \(y = 8\) at \(x = 1\), solve the differential equation \[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{(9x + 6)y^{\frac{1}{3}}}{x}\] giving your answer in the form \(y^2 = \mathrm{g}(x)\). (6)
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \frac{9x + 6}{x}\,\mathrm{d}x = \int \left(9 + \frac{6}{x}\right)\mathrm{d}x\) | M1 |
| \(= 9x + 6\ln x\quad (+C)\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \frac{1}{y^{\frac{1}{3}}}\,\mathrm{d}y = \int \frac{9x + 6}{x}\,\mathrm{d}x\) Integral signs not necessary | B1 |
| \(\displaystyle\int y^{-\frac{1}{3}}\,\mathrm{d}y = \int \frac{9x + 6}{x}\,\mathrm{d}x\) | |
| \(\dfrac{y^{\frac{2}{3}}}{\frac{2}{3}} = 9x + 6\ln x\quad (+C)\) \(\pm ky^{\frac{2}{3}} =\) their (a) | M1 |
| \(\dfrac{3}{2}y^{\frac{2}{3}} = 9x + 6\ln x\quad (+C)\) ft their (a) | A1ft |
| \(y = 8,\ x = 1\) \(\dfrac{3}{2}8^{\frac{2}{3}} = 9 + 6\ln 1 + C\) | M1 |
| \(C = -3\) | A1 |
| \(y^{\frac{2}{3}} = \dfrac{2}{3}(9x + 6\ln x - 3)\) | |
| \(y^2 = (6x + 4\ln x - 2)^3\quad \left(= 8(3x + 2\ln x - 1)^3\right)\) | A1 |
| (6) | |
| (8 marks) |