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  1. 6(a)(i)Obtain the general solution of the differential equation \cos x \frac{dy}{dx} + y\sin x = 2x\cos^2 x.[7 marks]
  2. 6(a)(ii)Hence, given that y = \frac{15\sqrt{2}\pi^2}{32} when x = \frac{\pi}{4}, determine the constant of the integration.[5 marks]
  3. 6(b)(i)a)The general solution of the differential equation y'' + 2y' + 5y = 4\sin 2t is y = CF + PI, where CF is the complementary function and PI is a…[2 marks]
  4. 6(b)(i)b)Hence, obtain the complementary function (CF), the general solution of y'' + 2y' + 5y = 0.[3 marks]
  5. 6(b)(ii)Given that the form of the particular integral (PI) is u_p(t) = A\cos 2t + B\sin 2t, show that A = -\frac{16}{17} and B = \frac{4}{17}.[3 marks]

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