FP2 June 2018 Q6
6.
(a) Find the general solution of the differential equation \[6\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 5\frac{\mathrm{d}y}{\mathrm{d}x} - 6y = x - 6x^2\] (8)
(b) Find the particular solution for which \(y = 0\) and \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3}{2}\) when \(x = 0\) (5)
| Scheme | Marks |
|---|---|
| \(6\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 5\dfrac{\mathrm{d}y}{\mathrm{d}x} - 6y = x - 6x^2\) | |
| \(6m^2 + 5m - 6 = 0 \Rightarrow (3m - 2)(2m + 3) = 0\) \(m = \dfrac{2}{3},\ \dfrac{-3}{2}\) M1 Forms and solves auxiliary equation A1 Correct roots | M1A1 |
| Complementary Function \(A\mathrm{e}^{\frac{2}{3}x} + B\mathrm{e}^{-\frac{3}{2}x}\) CF of the form shown formed using their 2 real roots Can be awarded if seen in gen solution | B1ft NB A1 on e-PEN |
| Particular Integral \((y =)\ Cx^2 + Dx + E\) May include higher powers | B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2Cx + D,\ \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 2C\) Differentiates their PI twice All powers of \(x\) to decrease by 1 | M1 |
| \(6(2C) + 5(2Cx + D) - 6\left(Cx^2 + Dx + E\right) \equiv -6x^2 + x\) | |
| \(-6C = -6\) \(10C - 6D = 1\) \(12C + 5D - 6E = 0\) Substitutes their derivatives into the equation and equates at least one pair of coefficients | M1 |
| \(C = 1\) \(10 - 6D = 1 \Rightarrow D = \dfrac{3}{2}\) \(12 + 5\left(\dfrac{3}{2}\right) - 6E = 0 \Rightarrow E = \dfrac{13}{4}\) Attempt to solve 3 equations. Must reach a numerical value for all 3 coefficients | M1 |
| General Solution \(y = A\mathrm{e}^{\frac{2}{3}x} + B\mathrm{e}^{-\frac{3}{2}x} + x^2 + \dfrac{3}{2}x + \dfrac{13}{4}\) Must start \(y = \ldots\) cao | A1 |
| (8) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2}{3}A\mathrm{e}^{\frac{2}{3}x} - \dfrac{3}{2}B\mathrm{e}^{-\frac{3}{2}x} + 2x + \dfrac{3}{2}\) Differentiates their GS – min 4 terms in their GS | M1 |
| \(y = 0,\ \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3}{2},\ x = 0 \qquad 0 = A + B + \dfrac{13}{4}\) \(\dfrac{3}{2} = \dfrac{2}{3}A - \dfrac{3}{2}B + \dfrac{3}{2}\) Forms 2 simultaneous equations using given boundary values | M1 |
| \(4A + 4B = -13,\ 4A - 9B = 0\) Attempt to solve Must reach \(A = \ldots\) or \(B = \ldots\) | M1 |
| \(A = -\dfrac{9}{4},\ B = -1\) Both correct | A1 |
| \(y = x^2 + \dfrac{3}{2}x + \dfrac{13}{4} - \dfrac{9}{4}\mathrm{e}^{\frac{2}{3}x} - \mathrm{e}^{-\frac{3}{2}x}\) Must start \(y = \ldots\) | A1 |
| (5) | |
| (13 marks) |