Finals(2090S18).pdf - Math 2090-3 Spring 2018 Final Exam...

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Math 2090-3 Spring 2018 Final Exam Name: Answer any 8 questions. To earn partial credit, you need to show all your work. 1. (a) Obtain the general solution to the separable differential equation dy dx = 2 x ( y 2 + 1) x 2 + 3 (b) Solve the linear first order differential equation y 0 - y x = 2 x 2 ln x, x > 0 . 1
2 2. Show that the equation (4e 2 x + 2 xy - y 2 ) dx + ( x - y ) 2 dy = 0 is exact, and hence solve it.
3 3. Determine all values of the constant k for which the system x 1 + 2 x 2 - x 3 = 3 , 2 x 1 + 5 x 2 + x 3 = 7 , x 1 + x 2 - k 2 x 3 = - k has (a) no solution, (b) an infinite number of solutions, (c) a unique solution.
4 4. Use the Gauss-Jordan technique to determine the inverse A - 1 , if possible, for the matrix 3 0 0 0 2 - 1 .
1 - 1 2
5 5. Use Cramer’s rule to determine the unique solution to the system x 1 + x 2 - 2 x 3 = - 2 x 2 + x 3 = 3 , 2 x 1 + 4 x 2 - 3 x 3 = 1 .
6 6. Determine all eigenvalues and corresponding eigenvectors of

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