Question 3 - A-Level Maths

Edexcel FP2 2005 June — Question 3 12 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2005
Session	June
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Second order differential equations
Type	Solve via substitution then back-substitute
Difficulty	Challenging +1.2 This is a structured FP2 differential equations question with clear guidance through substitution. Part (a) is algebraic verification requiring product rule and chain rule but following a given transformation. Parts (b) and (c) involve standard second-order DE techniques (complementary function + particular integral) with straightforward back-substitution. While requiring multiple calculus skills, the scaffolding and routine nature of each step place it moderately above average difficulty.
Spec	4.10a General/particular solutions: of differential equations 4.10e Second order non-homogeneous: complementary + particular integral

3. (a) Show that the transformation $y = x v$ transforms the equation $$\begin{array} { l l } x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 2 + 9 x ^ { 2 } \right) y = x ^ { 5 } , \\ \text { into the equation } & \frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 9 v = x ^ { 2 } . \end{array}$$I (b) Solve the differential equation II to find $v$ as a function of $x$.
(c) Hence state the general solution of the differential equation I.
(1)(Total 12 marks)

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Question 3:

Part (a):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$\frac{dy}{dx} = x\frac{dv}{dx} + v$	M1	For diff. product
$\frac{d^2y}{dx^2} = x\frac{d^2v}{dx^2} + 2\frac{dv}{dx}$	A1	Both correct
$x^2\left(x\frac{d^2v}{dx^2} + 2\frac{dv}{dx}\right) - 2x\left(x\frac{dv}{dx} + v\right) + (2+9x^2)vx = x^5$	M1	Substitution
$x^3\frac{d^2v}{dx^2} + 2x^2\frac{dv}{dx} - 2x^2\frac{dv}{dx} - 2vx + 2vx + 9vx^3 = x^5$	A1	Expansion
$\frac{d^2v}{dx^2} + 9v = x^2$	A1	cso

Part (b):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
CF: $v = A\sin 3x + b\cos 3x$	M1A1	May just write it down
Appropriate form for PI: $v = \lambda x^2 + \mu$ (or $ax^2 + bx + c$)	M1
Complete method to find $\lambda$ and $\mu$ (or $a, b, c$)	M1
$v = A\sin 3x + B\cos 3x + \frac{1}{9}x^2 - \frac{2}{81}$	M1A1ft6	f.t. only on wrong CF

Part (c):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$y = Ax\sin 3x + Bx\cos 3x + \frac{1}{9}x^3 - \frac{2}{81}x$	B1ft	f.t. for $y = x \times$ (candidate's CF + PI), providing two arbitrary constants

# Question 3:

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = x\frac{dv}{dx} + v$ | M1 | For diff. product |
| $\frac{d^2y}{dx^2} = x\frac{d^2v}{dx^2} + 2\frac{dv}{dx}$ | A1 | Both correct |
| $x^2\left(x\frac{d^2v}{dx^2} + 2\frac{dv}{dx}\right) - 2x\left(x\frac{dv}{dx} + v\right) + (2+9x^2)vx = x^5$ | M1 | Substitution |
| $x^3\frac{d^2v}{dx^2} + 2x^2\frac{dv}{dx} - 2x^2\frac{dv}{dx} - 2vx + 2vx + 9vx^3 = x^5$ | A1 | Expansion |
| $\frac{d^2v}{dx^2} + 9v = x^2$ | A1 | cso |

## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| CF: $v = A\sin 3x + b\cos 3x$ | M1A1 | May just write it down |
| Appropriate form for PI: $v = \lambda x^2 + \mu$ (or $ax^2 + bx + c$) | M1 | |
| Complete method to find $\lambda$ and $\mu$ (or $a, b, c$) | M1 | |
| $v = A\sin 3x + B\cos 3x + \frac{1}{9}x^2 - \frac{2}{81}$ | M1A1ft6 | f.t. only on wrong CF |

## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = Ax\sin 3x + Bx\cos 3x + \frac{1}{9}x^3 - \frac{2}{81}x$ | B1ft | f.t. for $y = x \times$ (candidate's CF + PI), providing two arbitrary constants |

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3. (a) Show that the transformation $y = x v$ transforms the equation

$$\begin{array} { l l } 
x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 2 + 9 x ^ { 2 } \right) y = x ^ { 5 } , \\
\text { into the equation } & \frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 9 v = x ^ { 2 } .
\end{array}$$I

(b) Solve the differential equation II to find $v$ as a function of $x$.\\
(c) Hence state the general solution of the differential equation I.\\
(1)(Total 12 marks)\\

\hfill \mbox{\textit{Edexcel FP2 2005 Q3 [12]}}

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