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MTH1030 -1035 - S1 2025 (Mission critical) Mock final exam (sample multiple choice and short answer questions)

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Question textThis is one of the problem sets that was part of the mock exam during COVID when the exam was fully online. There will be a problem set dedicated to differential equations on the real exam. However, that one will be hand-written. In general, to prepare for the differential equation part of the final exam, please make sure that you are familiar with the sample problems in the lecture notes and the applied class problem set.a) The differential equation with the initial condition y' = 2 x y, y(0) = 2has a solution of the form y(x) =What are c and d?Answer:c= Answer 1 Question 10[input]d= Answer 2 Question 10[input]b) A possible integrating factor for this differential equation y' + 3y/x = ex/x3is xc.What is c?Answer:c= Answer 3 Question 10[input]c) The differential equation y'' + 3 y' + 3 y = 6x+21has a particular solution of the form yp(x)=ax+b. What are a and b?Answer:a = Answer 4 Question 10[input]b = Answer 5 Question 10[input]

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Before diving into conclusions, let's unpack each subpart and walk through the reasoning step by step. Part a) The differential equation y' = 2 x y with initial condition y(0) = 2. - Approach: This is a separable first-order ODE. Rewrite as dy/y = 2x dx and integrate to obtain ln|y| = x^2 + C, so y = C e^{x^2}. - Determining the constant: Apply the initial condition y(0) = 2, which gives 2 = C e^{0} = C. Therefore, C = 2, and the solution is y(x) = 2 e^{x^2}. - If the problem expresses the solution in a form like y(x) = c e^{x^2} (with an extra parameter d present in the template), then c would be 2.......Login to view full explanation

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