practice-problems-final_2012 - M ATH 2090P RACTICE PROBLEMS...

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MATH 2090–P RACTICE PROBLEMS FOR THE FINAL EXAM N OVEMBER 2012 Use short, precise and complete English sentences to explain carefully your an- swers. Supply enough words so that the steps in your reasoning can be easily fol- lowed. (1) Solve the initial value problem dy dx = 2 xy 1 + x 2 ; y (0) = 3. (2) Find the general solution of the differential equation dy dx + tan( x ) y = cos( x ). (3) Solve the initial value problem y 0 = x 2 + 3 y 2 2 xy ; y (1) = 1. (4) Find the general solution of the Bernoulli differential equation y 0 - tan( x ) y = y 2 cos( x ). (5) Solve the system of linear equations 3 x 1 - x 2 + 3 x 3 - x 4 = 2 - x 1 + 5 x 2 - x 3 + 5 x 4 = 4 x 1 + 2 x 2 + x 3 + 2 x 4 = 3 - 2 x 2 - 2 x 4 = - 2 using Gauss-Jordan elimination (i.e. putting the associated augmented ma- trix in reduced row-echelon form). (6) Determine the value(s) of k for which the matrix below is the augmented ma- trix of a consistent linear system of equations.

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