Question 3 Consider the system of ODEs modelling the motion of a particle in R²: sin [8]-[-] [8] = [] t with (a) What is the solution to this ODE? (b) If we define the kinetic energy of the particle as E(t) = x(t)² + y(t)², show that E(t) is constant in time. (c) Suppose we use the explicit Euler method to solve this ODE on [0, 7] for a fixed timestep h = T/n for some n E N. If our approximate solution is 2 and y, then define the corresponding energy as Ek = x+y. Show that Ek+1 ≤ Ek+2√√2h√/Ek+h². Hint: recall ||*||1 ≤ √2||*||2 for all x R².
Question 3 Consider the system of ODEs modelling the motion of a particle in R²: sin [8]-[-] [8] = [] t with (a) What is the solution to this ODE? (b) If we define the kinetic energy of the particle as E(t) = x(t)² + y(t)², show that E(t) is constant in time. (c) Suppose we use the explicit Euler method to solve this ODE on [0, 7] for a fixed timestep h = T/n for some n E N. If our approximate solution is 2 and y, then define the corresponding energy as Ek = x+y. Show that Ek+1 ≤ Ek+2√√2h√/Ek+h². Hint: recall ||*||1 ≤ √2||*||2 for all x R².
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 3
Consider the system of ODEs modelling the motion of a particle in R²:
[8] - B
=
sin
[8]-[-],
=
t
with
(a) What is the solution to this ODE?
(b) If we define the kinetic energy of the particle as E(t) = x(t)² + y(t)², show that E (t) is constant
in time.
(c) Suppose we use the explicit Euler method to solve this ODE on [0, T] for a fixed timestep h = T/n
for some n € N. If our approximate solution is r and yk, then define the corresponding energy as
Ek = x+y Show that
Ek+1 ≤ Ek +2√2h√/Ek+h².
Hint: recall ||||1 ≤ √2||||2 for all x € R².
(d) Using (c), and assuming that Ek ≥ 1 for all k, show that at the final time,
²√2T - 1
2√2
Hint: if x ≥ 1 then √x≤x. Follow ideas from the analysis of the explicit Euler method.
En ≤e²√²T E +
T
n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b4ab23-f6d5-4206-a94e-63cad8267fd4%2F994c370f-15ad-4a5c-a6af-0279fdf42164%2Fwu6cks_processed.png&w=3840&q=75)
Transcribed Image Text:Question 3
Consider the system of ODEs modelling the motion of a particle in R²:
[8] - B
=
sin
[8]-[-],
=
t
with
(a) What is the solution to this ODE?
(b) If we define the kinetic energy of the particle as E(t) = x(t)² + y(t)², show that E (t) is constant
in time.
(c) Suppose we use the explicit Euler method to solve this ODE on [0, T] for a fixed timestep h = T/n
for some n € N. If our approximate solution is r and yk, then define the corresponding energy as
Ek = x+y Show that
Ek+1 ≤ Ek +2√2h√/Ek+h².
Hint: recall ||||1 ≤ √2||||2 for all x € R².
(d) Using (c), and assuming that Ek ≥ 1 for all k, show that at the final time,
²√2T - 1
2√2
Hint: if x ≥ 1 then √x≤x. Follow ideas from the analysis of the explicit Euler method.
En ≤e²√²T E +
T
n
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