Question: Question 3 (12 marks) a. Consider the following linear first-order differential equation dy + 2 y = e-* 1(0) = 2 dx 1 . By

Question 3 (12 marks) a. Consider the following linear first-order differential equation dy + 2 y = e-* 1(0) = 2 dx 1 . By comparing the equation to the standard form for a first-order linear differential dx equation ~+ P(x)y =O(x), identify P(x) and O(x). (2 marks) 11. Calculate the integrating factor v(x)=e P(x)ox (2 marks) 111. Solve the differential equation and use the initial value to find y( x) in terms of x . (3 marks) iv. State what happens to y ( x) as x -> co. (1 mark) b. Find the solution of the following separable differential equation, expressing y in terms of x dy -4xy y (1) =1 dx (4 marks)

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