midterm1 - A x = b : Do not continue to solve the system....

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Math 244 Midterm Exam 1 10/13/09 Mike Robertson Section: Name: Instructions. No notes or calculators allowed on this exam. Please turn o± your phones. This exam is worth the problem numbers. 1. (25) Show that the function provided is a solution to the di±erential equation y 0 + y = y 3 : Note that this equation is non-linear, and non-separable. y = 1 p 1 + 3 e 2 x 2. (5 ; 5) For the di±erential equation y 0 = x +2 y 2 x + y ; provide initial conditions of the form y ( a ) = b for which (a) there is guaranteed to be a unique solution. (b) there is no guarantee that there is a solution, or if a solution is found that it²s the only solution. 3. (25) Solve the following di±erential equation. y 0 5 x 1 y = x 5 ; y (1) = 3 4. (25) Solve the following di±erential equation. y 0 = e 3 x y 2 ; y (0) = 2 5. (15) Write the following system of equations as a matrix equation of the form
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Unformatted text preview: A x = b : Do not continue to solve the system. & 4 x & 2 y = 1 x + 3 y & z = 5 6. (20) Use only steps allowed by Gaussian Elimination to reduce the following matrix to Row Reduced Echelon Form (RREF). 2 4 1 2 & 3 2 4 & 2 2 3 6 & 4 3 3 5 7. (20) Solve for x in the following equation. 1 1 1 2 x = 1 & 1 2 1 8. (20) List &ve signi&cant statements which are logically equivalent to the statement " A has an inverse" 9. (20) Find the inverse for the matrix which represents 2 R 1 + R 2 ; & 1 R 1 + R 3 on a matrix with three rows. 10. (20) Use Cramers Rule to &nd values for a and g: Ten points of this will rely on you evaluating the determinant of 2 4 1 1 1 & 1 & 1 2 1 3 5 : 2 4 1 1 1 & 1 & 1 2 1 3 5 2 4 a b c d e f g h i j k l 3 5 = 2 4 3 & 3 1 5 1 1 2 1 1 3 5 1...
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