Section B For some a, 3 E R, let (6 3). 9e-t A = and b(t) = for all teR. %3D Be-t (a) Calculate an expression for a Fundamental Matrix 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 i = Ax with a(0) = ao. (c) Find the values of a for which the solution in (b) is bounded for all values of teR4 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 i = 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.
Section B For some a, 3 E R, let (6 3). 9e-t A = and b(t) = for all teR. %3D Be-t (a) Calculate an expression for a Fundamental Matrix 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 i = Ax with a(0) = ao. (c) Find the values of a for which the solution in (b) is bounded for all values of teR4 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 i = 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.
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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Question
part D E ,.,
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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