(a) Solve the linear system below using Laplace Transforms dX dy =X-aY dt for some real constant a, and X(0) = 0, X(0) = 1. dt = -aX + Y, (b) Find the range of values of a where (XY)=(0,0) is an unstable saddle. (c) If a = 0.1, what initial conditions x(0) > xo, y(0) > yo near (xo, Yo) will guarantee a faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the phase portrait!]
(a) Solve the linear system below using Laplace Transforms dX dy =X-aY dt for some real constant a, and X(0) = 0, X(0) = 1. dt = -aX + Y, (b) Find the range of values of a where (XY)=(0,0) is an unstable saddle. (c) If a = 0.1, what initial conditions x(0) > xo, y(0) > yo near (xo, Yo) will guarantee a faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the phase portrait!]
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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![The following dynamical systems model was recently proposed by Garcia and
Redondo (2022) "Dynamical systems approach in automobiles technological
transition from environmental drivers" to represent the technological transition of
automobiles from conventional (x) to electric (y):
dx
dt
= ax - 8xy,
dy
dt
The change over time of the number of automobiles of a given type is modelled as
the
115
= -1xy + By
difference between the automobiles of that type entering the market and coming out
of the
market per year. Phase portraits involving a saddle-node at (x, Y&), X&> 0, Y&> 0
were analysed to better understand scenarios where both technologies grow initially
but with one
dt
type growing much faster than the other (e.g., electric cars dominating over
conventional cars) with time.
(a) Solve the linear system below using Laplace Transforms
dX
dY
= -aX+Y,
= X - aY
dt
for some real constant a, and X(0) = 0, Y(0) = 1.
(b) Find the range of values of a where (X,Y)=(0,0) is an unstable saddle.
(c) If a = 0.1, what initial conditions x(0) > xo, y(0) > yo near (xo, yo) will guarantee a
faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the
phase portrait!]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2acbc3bd-375b-4deb-ac17-074e2cf36c97%2F99779fc3-7366-4504-9db7-9b0dcc3d6d7c%2F8q7tzwn_processed.png&w=3840&q=75)
Transcribed Image Text:The following dynamical systems model was recently proposed by Garcia and
Redondo (2022) "Dynamical systems approach in automobiles technological
transition from environmental drivers" to represent the technological transition of
automobiles from conventional (x) to electric (y):
dx
dt
= ax - 8xy,
dy
dt
The change over time of the number of automobiles of a given type is modelled as
the
115
= -1xy + By
difference between the automobiles of that type entering the market and coming out
of the
market per year. Phase portraits involving a saddle-node at (x, Y&), X&> 0, Y&> 0
were analysed to better understand scenarios where both technologies grow initially
but with one
dt
type growing much faster than the other (e.g., electric cars dominating over
conventional cars) with time.
(a) Solve the linear system below using Laplace Transforms
dX
dY
= -aX+Y,
= X - aY
dt
for some real constant a, and X(0) = 0, Y(0) = 1.
(b) Find the range of values of a where (X,Y)=(0,0) is an unstable saddle.
(c) If a = 0.1, what initial conditions x(0) > xo, y(0) > yo near (xo, yo) will guarantee a
faster rate of consumer uptake of electric cars than conventional cars? [Hint: Plot the
phase portrait!]
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