Spring 08 exam 3 soln

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Mathematics 38 Differential Equations Examination 3 April 7, 12:001:20 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....
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online