Difficult-to-solve differential equations can always be approximated by numerical methods. We look at one numerical method called Euler's Method. Euler's method uses the readily available slope information to start from the point (xo, yo) then move from one point to the next along the polygon approximation of the graph of the particular differential equation to ultimately reach the terminal point, (x,y). Al- though interested in determining all of the points along the differential equation, it is often the case that the value of y, at the terminal point is of most interest. More specifically, let y = f(x) be the solution to the differential equation = g(x,y), with y(x) = yo For Xo ≤x≤x, and let X1+1=x+h, where h = -x and 12 Yi+1=Y₁+ g(x,y)h For 0≤i≤n-1, then f(x1+1) ≈Yi+1 g(x,y)=... 1 1. Approximate: x = y - 2x start at (x, y) = (0,1), 0 ≤ x ≤2, when h = 0.4 Since dy = y - 2x then dx (i) y - 2x (iii) 3e¹/2 (ii) 2x+2-ex And since x = 0, yo = 1 then g(x,yo) yo - 2x₁ = 1-2(0)= (i) 0 (ii) 1 (iii) 2, And since h = 0.4, (i) 1 Y₁ yo+g(xo, yo)h = 1 + 1(0.4)= But now, since x₁ = x0 + h = 0 + 0.4 0.4 and g(x₁, y₁) = y₁- 2x₁ = 1.4 - 2(0.4)= (ii) 1.4 (iii) 1.8, (i) 0.6 And since h = 0.4, (ii) 0.8 (iii) 1.0, (i) 1.64 y2y+g(x1,y₁)h = 1.4 + 0.6(0.4) = Complete the following table: i Euler's Approximation (ii) 1.84 (iii) 2.04 Actual Solution Difference y-f(x) 0 1 2 3 4 5 a) Repeat question 1 with h = 0.1 b) Euler's method improves or worsens for smaller h subintervals. 3 Approximate bear population assuming the bear population grows according to following differential equation: dy at=y(y+1)(y+3) Assume initial population at time t = 0 is y = 5, use Euler's method, where h = 1 year, to approximate bear population at time t = 3 years. NB: Follow the same steps as question one
Difficult-to-solve differential equations can always be approximated by numerical methods. We look at one numerical method called Euler's Method. Euler's method uses the readily available slope information to start from the point (xo, yo) then move from one point to the next along the polygon approximation of the graph of the particular differential equation to ultimately reach the terminal point, (x,y). Al- though interested in determining all of the points along the differential equation, it is often the case that the value of y, at the terminal point is of most interest. More specifically, let y = f(x) be the solution to the differential equation = g(x,y), with y(x) = yo For Xo ≤x≤x, and let X1+1=x+h, where h = -x and 12 Yi+1=Y₁+ g(x,y)h For 0≤i≤n-1, then f(x1+1) ≈Yi+1 g(x,y)=... 1 1. Approximate: x = y - 2x start at (x, y) = (0,1), 0 ≤ x ≤2, when h = 0.4 Since dy = y - 2x then dx (i) y - 2x (iii) 3e¹/2 (ii) 2x+2-ex And since x = 0, yo = 1 then g(x,yo) yo - 2x₁ = 1-2(0)= (i) 0 (ii) 1 (iii) 2, And since h = 0.4, (i) 1 Y₁ yo+g(xo, yo)h = 1 + 1(0.4)= But now, since x₁ = x0 + h = 0 + 0.4 0.4 and g(x₁, y₁) = y₁- 2x₁ = 1.4 - 2(0.4)= (ii) 1.4 (iii) 1.8, (i) 0.6 And since h = 0.4, (ii) 0.8 (iii) 1.0, (i) 1.64 y2y+g(x1,y₁)h = 1.4 + 0.6(0.4) = Complete the following table: i Euler's Approximation (ii) 1.84 (iii) 2.04 Actual Solution Difference y-f(x) 0 1 2 3 4 5 a) Repeat question 1 with h = 0.1 b) Euler's method improves or worsens for smaller h subintervals. 3 Approximate bear population assuming the bear population grows according to following differential equation: dy at=y(y+1)(y+3) Assume initial population at time t = 0 is y = 5, use Euler's method, where h = 1 year, to approximate bear population at time t = 3 years. NB: Follow the same steps as question one
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
Please help me with this homework, NOTE: this is physics not mathematics.
Transcribed Image Text:Difficult-to-solve differential equations can always be approximated by numerical
methods. We look at one numerical method called Euler's Method. Euler's method uses
the readily available slope information to start from the point (xo, yo) then move from one
point to the next along the polygon approximation of the graph of the particular differential
equation to ultimately reach the terminal point, (x,y). Al- though interested in
determining all of the points along the differential equation, it is often the case that the
value of y, at the terminal point is of most interest. More specifically, let y = f(x) be
the solution to the differential equation
= g(x,y), with y(x) = yo
For Xo ≤x≤x, and let
X1+1=x+h, where
h = -x and
12
Yi+1=Y₁+ g(x,y)h
For 0≤i≤n-1, then
f(x1+1)
≈Yi+1
g(x,y)=...
1
1. Approximate: x = y - 2x start at (x, y) = (0,1), 0 ≤ x ≤2, when h = 0.4
Since dy = y - 2x then
dx
(i) y - 2x
(iii) 3e¹/2
(ii) 2x+2-ex
And since x = 0, yo = 1 then
g(x,yo) yo - 2x₁ = 1-2(0)=
(i) 0
(ii) 1
(iii) 2,
And since h = 0.4,
(i) 1
Y₁ yo+g(xo, yo)h = 1 + 1(0.4)=
But now, since x₁ = x0 + h = 0 + 0.4 0.4 and
g(x₁, y₁) = y₁- 2x₁ = 1.4 - 2(0.4)=
(ii) 1.4
(iii) 1.8,
(i) 0.6
And since h = 0.4,
(ii) 0.8
(iii) 1.0,
(i) 1.64
y2y+g(x1,y₁)h = 1.4 + 0.6(0.4) =
Complete the following table:
i
Euler's
Approximation
(ii) 1.84
(iii) 2.04
Actual Solution
Difference y-f(x)
0
1
2
3
4
5
Transcribed Image Text:a) Repeat question 1 with h = 0.1
b) Euler's method improves or worsens for smaller h subintervals.
3
Approximate bear population assuming the bear population grows according to following
differential equation:
dy
at=y(y+1)(y+3)
Assume initial population at time t = 0 is y = 5, use Euler's method, where h = 1 year, to
approximate bear population at time t = 3 years.
NB: Follow the same steps as question one
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON