2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. b) consider the system: (1) A i) if y = 0 then -t Show that the following vector function () = (a+bt) e- is a solution of (1) if and only if c+dt. = (2)) = - (²2) c) use the above exercises to find the general solution d) Asses whether the solution you got consist of two linearly independent functions. e) explain why the solution curves, when t → +∞o are parallel with x-axis f) find the solution curve through (x(to), y(to)) = (xo, 0), for a given x。 * 0 g) find the tangent direction of the solution curves when intersecting the y-axis of two linearly independent functions. h) use the chain rule to show that dx x-5y dy y A-(1¹5) 0 = А 0 along the line x = = = -x + 5y y' = -y dx dy j) make a drawing of the solution curve in the phase plane k) what type of critical point is the origin and determine the stability of this. I) consider the curves given by: and ^ ( )= (-a) A () = ( 5y. How do you interpret this in the phase plane? (C₁+C₂t)e- e-t 5y - and let (x(0), y(0)) = (0,2). Show that x(t) = ½ty and t = − In (52) for 5 > 0. C₂
2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. b) consider the system: (1) A i) if y = 0 then -t Show that the following vector function () = (a+bt) e- is a solution of (1) if and only if c+dt. = (2)) = - (²2) c) use the above exercises to find the general solution d) Asses whether the solution you got consist of two linearly independent functions. e) explain why the solution curves, when t → +∞o are parallel with x-axis f) find the solution curve through (x(to), y(to)) = (xo, 0), for a given x。 * 0 g) find the tangent direction of the solution curves when intersecting the y-axis of two linearly independent functions. h) use the chain rule to show that dx x-5y dy y A-(1¹5) 0 = А 0 along the line x = = = -x + 5y y' = -y dx dy j) make a drawing of the solution curve in the phase plane k) what type of critical point is the origin and determine the stability of this. I) consider the curves given by: and ^ ( )= (-a) A () = ( 5y. How do you interpret this in the phase plane? (C₁+C₂t)e- e-t 5y - and let (x(0), y(0)) = (0,2). Show that x(t) = ½ty and t = − In (52) for 5 > 0. C₂
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
2g
Transcribed Image Text:2.-
a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A.
A - )
5
b) consider the system:
(x' = -x + 5y
y' = -y
(1)
Show that the following vector function
(*DE)e-t is a solution of (1) if and only if
.c+dt
A (") - C-)
A
and
= -
c) use the above exercises to find the general solution
d) Asses whether the solution you got consist of two linearly independent functions.
e) explain why the solution curves, when t → t0 are parallel with x-axis
f) find the solution curve through (x(to), y(to)) = (xo,0), for a given x, + 0
g) find the tangent direction of the solution curves when intersecting the y-axis of two linearly
independent functions.
dx
x-5y
=
h) use the chain rule to show that-
dy
у
dx
i) if y + 0 then
dy
O along the line x
5y. How do you interpret this in the phase plane?
j) make a drawing of the solution curve in the phase plane
k) what type of critical point is the origin and determine the stability of this.
I) consider the curves given by:
G)-(*
(c1 + c2t)e
5y
and let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In
1(2) for> 0.
C2
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