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For some a, 3 e R, let -(; :). () 9e-t A = and b(t) = (t) for all te R4. Be-t (a) Calculate an expression for a Fundamental Matriz Function : R. C2x2 associated with A. (b) Use the calculated Fundamental Matrix Function to find the solution, in terms of a, to the homogeneous ODE * = Ax with r(0) = r0. %3D (c) Find the values of a for which the solution in (b) is bounded for all values of t ER+ and hence give an expression for those solutions. Van Juti

For some a, 3 e R, let -(; :). () 9e-t A = and b(t) = (t) for all te R4. Be-t (a) Calculate an expression for a Fundamental Matriz Function : R. C2x2 associated with A. (b) Use the calculated Fundamental Matrix Function to find the solution, in terms of a, to the homogeneous ODE * = Ax with r(0) = r0. %3D (c) Find the values of a for which the solution in (b) is bounded for all values of t ER+ and hence give an expression for those solutions. Van Juti

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Section B For some a, 3 E R, let (6 3). () 9e-t Be-t A = and b(t) = for all teR. %3D (a) Calculate an expression for a Fundamental Matrix Function R associated with A. →C2×2 (b) Use the calculated Fundamental Matrix Function to find the solution, in terms of a, to the homogeneous ODE i = Aæ with a(0) = ao. (c) Find the values of a for which the solution in (b) is bounded for all values of t e R. and hence give an expression for those solutions. (d) Use the Variation of Parameters Formula to find the solution, in terms of a and 3, to the inhomogeneous ODE * = Aæ + b with (0) ao. (e) Determine a relationship between a and 3 for which the solution in (d) is bounded for all values of t E R4.
Transcribed Image Text:Section B For some a, 3 E R, let (6 3). () 9e-t Be-t A = and b(t) = for all teR. %3D (a) Calculate an expression for a Fundamental Matrix Function R associated with A. →C2×2 (b) Use the calculated Fundamental Matrix Function to find the solution, in terms of a, to the homogeneous ODE i = Aæ with a(0) = ao. (c) Find the values of a for which the solution in (b) is bounded for all values of t e R. and hence give an expression for those solutions. (d) Use the Variation of Parameters Formula to find the solution, in terms of a and 3, to the inhomogeneous ODE * = Aæ + b with (0) ao. (e) Determine a relationship between a and 3 for which the solution in (d) is bounded for all values of t E R4.
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