2 1 Let A = and consider the Euler equation tx' = Ax fort > 0. 0 2 (a) Show that the above system can be written as tx = 2x1 + x2 ta, = 2x2 (b) Show that t-2 is an integrating factor for the DE x' = 2t¬1x+t-!g(t), where g(t) is a continuous real function defined for t > 0. (c) Solve the DEs in part (a). Solve the second equation first. (d) Write down a fundamental matrix for the system tæ' = Ax for t > 0.
2 1 Let A = and consider the Euler equation tx' = Ax fort > 0. 0 2 (a) Show that the above system can be written as tx = 2x1 + x2 ta, = 2x2 (b) Show that t-2 is an integrating factor for the DE x' = 2t¬1x+t-!g(t), where g(t) is a continuous real function defined for t > 0. (c) Solve the DEs in part (a). Solve the second equation first. (d) Write down a fundamental matrix for the system tæ' = Ax for t > 0.
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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From part b) onwards please
Transcribed Image Text:2 1
Let A =
and consider the Euler equation tx' = Ax for t > 0.
0 2
(a) Show that the above system can be written as
tx = 2x1 + x2
ta', = 2x2
(b) Show that t-2 is an integrating factor for the DE x' = 2t-lx+t=1g(t), where
g(t) is a continuous real function defined for t > 0.
(c) Solve the DEs in part (a). Solve the second equation first.
(d) Write down a fundamental matrix for the system tx' = Ax for t > 0.
(e) Compute the state transition matrix d(t, 1) and use it to solve the initial value
2.
problem tx' = Ax, x(1) =
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