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a) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. Taking as the initial values y(0) = a, and y'(0) = b, (a,b pair assigned) solve the homogeneous equation (set equal to 0) by Laplace Transform.

a) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. Taking as the initial values y(0) = a, and y'(0) = b, (a,b pair assigned) solve the homogeneous equation (set equal to 0) by Laplace Transform.

Advanced Engineering Mathematics
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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a=-2 and b=1
a) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. Taking as the initial values y(0) = a, and y'(0) = b, (a, b pair assigned) solve the homogeneous equation (set equal to 0) by Laplace Transform. b) Using a) above, solve for the right hand side (bt+a)et with the initial values y(0) = b and y'(0) = 0 (b assigned) by Laplace Transform. c) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. This DE must be different from a) above. Form the following right-hand-side using your a, b pair : g(t) = { a bt g(t) = 0 ≤ t <3 t≥ 3 Solve this using Laplace Transform using the initial values y(0) = 1 and y'(0) = 0. d) For the differential equation you constructed in c) above, form the following right-hand-side using your a, b pair: a at² + b bt + a 0 ≤t <2 2 <t t> 4 Solve this using Laplace Transform using the initial values y(0) = 1 and y'(0) = = 0.
Transcribed Image Text:a) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. Taking as the initial values y(0) = a, and y'(0) = b, (a, b pair assigned) solve the homogeneous equation (set equal to 0) by Laplace Transform. b) Using a) above, solve for the right hand side (bt+a)et with the initial values y(0) = b and y'(0) = 0 (b assigned) by Laplace Transform. c) Construct a 2nd order differential equation with constant coefficients whose characteristic equation has unique roots neither of which are 0. This DE must be different from a) above. Form the following right-hand-side using your a, b pair : g(t) = { a bt g(t) = 0 ≤ t <3 t≥ 3 Solve this using Laplace Transform using the initial values y(0) = 1 and y'(0) = 0. d) For the differential equation you constructed in c) above, form the following right-hand-side using your a, b pair: a at² + b bt + a 0 ≤t <2 2 <t t> 4 Solve this using Laplace Transform using the initial values y(0) = 1 and y'(0) = = 0.
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