351sample_ex - M351 Sample Hour Examination Warning This...

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M351 Sample Hour Examination Warning: This set of questions is similar to an examination given in a previous year. There is NO GUARANTEE that the questions here will be in any way related to the questions that appear on the actual hour exam for the course OR that the coverage of topics will be the same. It is intended ONLY for practice. All problems are worth 20 points. 1. Consider the first order non-homogeneous equation dx dt - 1 t x = t 2 , t > 0 . (a) Write down the associated homogeneous equation. (b) Find the general solution of the homogeneous equation OR propose such a general solution and show that your choice indeed is a solution. (c) Show that the function t 3 2 is a solution of the original non-homogeneous dif- ferential equation and write down a general solution of the original equation using the result of part (b). 2. Find all constant solutions of the differential equation dy dx = y 2 - y - 2 , and then find an implicit solution satisfying the initial condition y (0) = 0.
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