The function \(y = \mathrm{f}(x)\) is such that \(\frac{\mathrm{d}y}{\mathrm{d}x} = y^2 \sin 3x\), and \(y = 1\) when \(x = \pi\).
Show that \(\mathrm{f}(x) = \frac{A}{B + \cos 3x}\), where \(A\) and \(B\) are constants to be determined.[4]
Sketch the graph of \(y = \mathrm{f}(x)\) for \(0 \leqslant x \leqslant 2\pi\). State the values of \(c\) for which the line \(y = c\) is a tangent to the curve.[3]
State the equations of the axes of symmetry of \(y = \mathrm{f}(x)\).[2]
Find an expression for \(\mathrm{f}(x + \pi)\), giving your answer in terms of \(\cos 3x\).[1]
Show that \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = Py^2 + Qy^3\), where \(P\) and \(Q\) are functions of \(x\) to be determined.[3]