AQA
Further Paper 2
2022
June
— Question 9
3 marks
Exam Board
AQA
Module
Further Paper 2 (Further Paper 2)
Year
2022
Session
June
Marks
3
Topic
First order differential equations (integrating factor)
9
A curve passes through the point (5, 12.3) and satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad x > 3$$
Use Euler's step by step method once, and then the midpoint formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right) , \quad x _ { r + 1 } = x _ { r } + h$$
once, each with a step length of 0.1 , to estimate the value of \(y\) when \(x = 5.2\)
Give your answer to six significant figures.
9
Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad ( x > 3 )$$
9
(ii) Given that \(y\) satisfies the differential equation in part (b)(i) and that \(y = 12.3\) when \(x = 5\), find the value of \(y\) when \(x = 5.2\)
Give your answer to six significant figures. [0pt]
[3 marks]
9
Comment on the accuracy of your answer to part (a).