Standard +0.8 This is a standard second-order linear differential equation with constant coefficients and exponential RHS from Further Maths FP3. While the method is routine (find complementary function via auxiliary equation, then particular integral), it requires multiple techniques and careful algebra. The exponential RHS (e^{3x}) doesn't match either root of the auxiliary equation (2 and 4), making it straightforward but still requiring competent execution across several steps—moderately above average difficulty.
3 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = \mathrm { e } ^ { 3 x } .$$
For GS, f.t. from their CF + PI with 2 arbitrary constants in CF and none in PI
**Aux. equation** $m^2 - 6m + 8 = (0)$ | M1 | For auxiliary equation seen
$m = 2, 4$ | A1 | For correct roots
**CF** $y = Ae^{2x} + Be^{4x}$ | A1√ | For correct CF, f.t. from their $m$
**PI** $(y =) Ce^{3x}$ | M1 | For stating and substituting PI of correct form
$9C - 18C + 8C = 1 \Rightarrow C = -1$ | A1 | For correct value of $C$
**GS** $y = Ae^{2x} + Be^{4x} - e^{3x}$ | B1√ 6 | For GS, f.t. from their CF + PI with 2 arbitrary constants in CF and none in PI