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# ps8[1] - = b 2 √ ac cos jπ n 1,j = 1 2,n(a Verify this...

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Applied Mathematics E2101y Introduction to Applied Mathematics Problem Set #8, due Tuesday, 28 April 2009 All references are to Zill & Cullen (3rd edition). Chapter 3, § 3.1: Higher-order ODEs 1, 5, 12, 13, 21, 22, 40 Chapter 3, § 3.2: Reduction of Order 5 (simplify the result to something you recognize!) 8 17 (note that a particular inhomogeneous solution is y = - 1 / 2) Chapter 3, § 3.3: Constant-coefficient Linear ODEs 11 15 30 40 43–48 (Multiple choice, but explain reasoning by inspecting a relevant quadratic equation.) 58 Chapter 3, § 3.4: Undetermined Coefficients 9 15 31 Supplementary Problem: Tridiagonal matrices with constant diagonals The n × n tridiagonal matrix A = b c 0 0 · · · 0 a b c 0 · · · 0 0 a b c · · · 0 . . . . . . . . . . . . . . . 0 0 · · · a b c 0 0 · · · 0 a b has eigenvalues λ j = b + 2
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Unformatted text preview: = b + 2 √ ac cos jπ n + 1 ,j = 1 , 2 ,...,n. (a) Verify this for n = 1 and then for n = 2 by direct calculation. (b) Verify this for n = 4 using MATLAB’s eig function, with a =-1, b = 2, and c =-1. (c) Verify this for general n but with c = a (symmetric case) by using the fact that the n × n matrix of eigenvectors, V , has elements v ij = sin ijπ n +1 . [Hint: write out a general interior row of A v j = λ j v j , which involves just three terms (since the matrix is tridiagonal) and use the trigonometric identity sin( α ± β ) = sin α cos β ± cos α sin β to replace the two oﬀ-diagonal terms (the ones involving a and c ) with multiples of the diagonal term.]...
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