Question: QUESTION 1 (20 MARKS) a) Consider the first-order differential equation (y + cos x) dx + dy = 0. By multiplying integrating factor u(x)

QUESTION 1 (20 MARKS) a) Consider the first-order differential equation (y +

QUESTION 1 (20 MARKS) a) Consider the first-order differential equation (y + cos x) dx + dy = 0. By multiplying integrating factor u(x) = e to both sides, show that the differential equation is exact. Hence, solve the differential equation. (6 marks) b) Solve the differential equation dy 2 + ytan x = dx (4x + 5) -y. COS X = SSCE 1793 c) The rate of the radioactive decay is governed by the equation dM d.t -kM, 2 (7 marks) where k is a constant and M is the amount of remaining radioactive material at time t. (i) Show that the solution of the above equation is M(t) = Ae-kt where A is a constant. (2 marks) (ii) The initial amount of radioactive material is 5000 and after 20 years 98.85% are left from the initial amount. Find the amount of radioactive material after 100 years. (5 marks)

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