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Question 2 Difference equations, Laplace transforms and the Calculus of Variations. a) Suppose we have the following 1st order difference equation yt = -1.2yt-1 + 198, find the particular solution using any method of your choice. Take yo = 50. b) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order difference equation (where yt is national income): Yt+2 - 5yt+1 + 6yt = 4t, if yt = 0 for t < 0, and yo = 0, y₁ = 1 1-e-s You may use without proof that L-¹[; [s(1-re-s)] = f(t) = r¹ for n ≤ t < n + 1.

Question 2 Difference equations, Laplace transforms and the Calculus of Variations. a) Suppose we have the following 1st order difference equation yt = -1.2yt-1 + 198, find the particular solution using any method of your choice. Take yo = 50. b) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order difference equation (where yt is national income): Yt+2 - 5yt+1 + 6yt = 4t, if yt = 0 for t < 0, and yo = 0, y₁ = 1 1-e-s You may use without proof that L-¹[; [s(1-re-s)] = f(t) = r¹ for n ≤ t < n + 1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 40E
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Question 2 Difference equations, Laplace transforms and the Calculus of Variations. a) Suppose we have the following 1st order difference equation yt = −1.2yt-1 + 198, find the particular solution using any method of your choice. Take yo = 50. b) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order difference equation (where yt is national income): Yt+2 - 5yt+1 + 6y₁ = 4t, if y₁ = 0 for t < 0, and yo = 0, y₁ = 1 1-e-s You may use without proof that L¹[(1-re-sj] = f(t) = r² for n ≤t<n+1. c) Using the Euler-Lagrange equation of the Calculus of Variations Optimize (-16y² + 144y + 11yy' − 4(y')²)dx Subject to y(0) = 8 and y(1) = 8.6, 0 < y(x) < 1
Transcribed Image Text:Question 2 Difference equations, Laplace transforms and the Calculus of Variations. a) Suppose we have the following 1st order difference equation yt = −1.2yt-1 + 198, find the particular solution using any method of your choice. Take yo = 50. b) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order difference equation (where yt is national income): Yt+2 - 5yt+1 + 6y₁ = 4t, if y₁ = 0 for t < 0, and yo = 0, y₁ = 1 1-e-s You may use without proof that L¹[(1-re-sj] = f(t) = r² for n ≤t<n+1. c) Using the Euler-Lagrange equation of the Calculus of Variations Optimize (-16y² + 144y + 11yy' − 4(y')²)dx Subject to y(0) = 8 and y(1) = 8.6, 0 < y(x) < 1
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