Standard +0.3 This is a standard second-order linear differential equation with constant coefficients and exponential RHS. Students follow a routine procedure: find the complementary function by solving the auxiliary equation, then find a particular integral (requiring adjustment since e^{3x} is part of the CF). While it requires careful algebraic manipulation and knowledge of the resonance case, it's a textbook exercise with well-established methods, making it slightly easier than average for Further Maths students.
Correct value of \(a\) (and any other coefficients as zero); must have had a suitable PI
General solution: \(y = Ae^{-\frac{1}{2}x} + Be^{3x} + \frac{2}{7}xe^{3x}\)
B1ft
\(y =\) their CF \(+\) their PI; must include \(y=\); PI must match initial choice with constants substituted
# Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2m^2 - 5m - 3 = 0 \Rightarrow (2m+1)(m-3) = 0 \Rightarrow m = \ldots$ | M1 | Forms and solves the auxiliary equation |
| C.F. is $y_{CF} = Ae^{-\frac{1}{2}x} + Be^{3x}$ | A1 | Correct complementary function (no need for $y=\ldots$) |
| P.I. is $y_{PI} = axe^{3x}$ | B1 | Correct form for P.I.; accept any PI that includes $axe^{3x}$, e.g. $(ax+b)e^{3x}$ is fine |
| $\frac{dy_{PI}}{dx} = 3axe^{3x} + ae^{3x},\ \frac{d^2y_{PI}}{dx^2} = 9axe^{3x} + 3ae^{3x} + 3ae^{3x}$ | M1 | Attempts to differentiate PI twice and substitutes into LHS; derivatives must be changed functions |
| $\Rightarrow 2(9ax+6a)e^{3x} - 5(3ax+a)e^{3x} - 3axe^{3x} = 2e^{3x} \Rightarrow a = \ldots$ | | |
| $a = \frac{2}{7}$ | A1 | Correct value of $a$ (and any other coefficients as zero); must have had a suitable PI |
| General solution: $y = Ae^{-\frac{1}{2}x} + Be^{3x} + \frac{2}{7}xe^{3x}$ | B1ft | $y =$ their CF $+$ their PI; must include $y=$; PI must match initial choice with constants substituted |
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