Standard +0.8 This is a standard integrating factor problem from Further Maths, requiring identification of the integrating factor (sin x), integration of sin 2x ยท sin x (requiring product-to-sum formula or double angle manipulation), and application of initial conditions. While methodical, it involves multiple non-trivial integration steps and trigonometric manipulation beyond typical A-level Core content, placing it moderately above average difficulty.
4 By using an integrating factor, find the solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$
given that \(y = \frac { 1 } { 2 }\) when \(x = \frac { \pi } { 6 }\).
(10 marks)
4 By using an integrating factor, find the solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$
given that $y = \frac { 1 } { 2 }$ when $x = \frac { \pi } { 6 }$.\\
(10 marks)
\hfill \mbox{\textit{AQA FP3 2011 Q4 [10]}}