Spring 08 exam 3 soln

# Spring 08 exam 3 soln - Mathematics 38 Differential...

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Unformatted text preview: Mathematics 38 Differential Equations Examination 3 April 7, 12:00–1:20 1. (10 points) In parts a. and b. you are given a matrix A , a vector-valued function vector E ( t ) and formulas describing a collection of solutions of the nonhomogeneous system Dvectorx = Avectorx + vector E ( t ) . In each case decide whether the collection is complete. a. A = parenleftbigg − 3 − 2 1 parenrightbigg , vector E ( t ) = parenleftbigg 2 e − t − e − t parenrightbigg : braceleftbigg x 1 = 2 c 1 e − 2 t + c 2 e − t x 2 = − c 1 e − 2 t − c 2 e − t + e − t . Solution: The Wronskian at 0 is det parenleftbigg 2 1 − 1 − 1 parenrightbigg = − 1 negationslash = 0 , so this is a complete set. b. A = 5 − 3 3 − 5 1 2 , vector E ( t ) = 4 : x 1 = 6 c 1 e 4 t − 2 c 2 e − 4 t x 2 = 2 c 1 e 4 t − 6 c 2 e − 4 t x 3 = c 1 e 4 t + c 2 e − 4 t − 2 . Solution: Since this is a generic linear combination of only 2 (not 3) solutions, this set is not complete. 2. (10 points) Check the following set of vectors for linear independence: 1 2 3 4 1 2 1 4 3 2 − 1 1 − 1 1 − 1 . Solution: Note that 1 2 3 4 1 − 2 1 4 3 2 − − 1 1 − 1 1 − 1 = vector , so this triple is linearly dependent. Alternatively, reduce 1 2 − 1 2 1 1 3 4 − 1 4 3 1 1 2 − 1 to 1 1 1 − 1 and note the presence of a free variable, which gives the solution − 1 , 1, 1. 3. (5 points) The matrix 1 4 1 3 3 7 − 3 6 1 3 1 9 − 5 8 5 6 has 1 1 as an eigenvector. Find the corresponding eigenvalue....
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Spring 08 exam 3 soln - Mathematics 38 Differential...

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