Pre-U Pre-U 9795/1 2011 June — Question 5 7 marks

Exam BoardPre-U
ModulePre-U 9795/1 (Pre-U Further Mathematics Paper 1)
Year2011
SessionJune
Marks7
TopicSecond order differential equations
TypeStandard non-homogeneous with polynomial RHS
DifficultyStandard +0.8 This is a second-order linear non-homogeneous differential equation requiring students to find both the complementary function (standard auxiliary equation giving sin/cos solutions) and a particular integral (requiring trial of a quadratic form). While the method is systematic, it involves multiple steps including differentiation of the trial solution twice, coefficient matching, and combining solutions—more demanding than routine single-technique questions but still a standard textbook problem type for Further Maths students.
Spec4.10e Second order non-homogeneous: complementary + particular integral
Find the general solution of the differential equation \(\frac{d^2 y}{dx^2} + y = 8x^2\). [7]
AnswerMarks
Aux. Eqn. \(m^2 + 1 = 0 \Rightarrow m = \pm i\) ⟹ Comp. Fn. is \(y_c = A\cos x + B\sin x\)M1, A1, M1
For Part. Intgrl. trying \(y = ax^2 + bx + c\) (with at least \(a\) non-zero)M1
\(\frac{dy}{dx} = 2ax + b\), \(\frac{d^2y}{dx^2} = 2a\)M1
Diff'. their \(y_p\) to find \(y'\) and \(y''\) and subst'. into the given d.e.M1
Equating terms to find \(a, b, c\) (with at least \(a\) and \(c\) non-zero)M1
\(a = 8, b = 0, c = -16\); i.e. \(y_p = 8x^2 - 16\)A1
Gen. Soln. is \(y = A\cos x + B\sin x + 8x^2 - 16\) ft their \(y_c + y_p\) provided \(y_c\) has 2 arb. consts. and \(y_p\) has noneB1
[7] 
Aux. Eqn. $m^2 + 1 = 0 \Rightarrow m = \pm i$ ⟹ Comp. Fn. is $y_c = A\cos x + B\sin x$ | M1, A1, M1 |
For Part. Intgrl. trying $y = ax^2 + bx + c$ (with at least $a$ non-zero) | M1 |
$\frac{dy}{dx} = 2ax + b$, $\frac{d^2y}{dx^2} = 2a$ | M1 |
Diff'. their $y_p$ to find $y'$ and $y''$ and subst'. into the given d.e. | M1 |
Equating terms to find $a, b, c$ (with at least $a$ and $c$ non-zero) | M1 |
$a = 8, b = 0, c = -16$; i.e. $y_p = 8x^2 - 16$ | A1 |
Gen. Soln. is $y = A\cos x + B\sin x + 8x^2 - 16$ ft their $y_c + y_p$ provided $y_c$ has 2 arb. consts. and $y_p$ has none | B1 |
| [7] |

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Find the general solution of the differential equation $\frac{d^2 y}{dx^2} + y = 8x^2$. [7]

\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2011 Q5 [7]}}
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