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Problem 3: Consider the IVP y" + ay + by = et, _y(0) = 1, y'(0) = 1. 1. Write the second-order IVP as an IVP of a first-order system of equations by setting y = z. Show that the eigenvalue of the resulting matrix are the same as the roots of the characteristic polynomial that is associated to the second-order IVP. 2. For the remainder of this question, set a = -1 and b = -6. 3. Find the exponential of the matrix of the system by using the diagonalization approach.

Problem 3: Consider the IVP y" + ay + by = et, _y(0) = 1, y'(0) = 1. 1. Write the second-order IVP as an IVP of a first-order system of equations by setting y = z. Show that the eigenvalue of the resulting matrix are the same as the roots of the characteristic polynomial that is associated to the second-order IVP. 2. For the remainder of this question, set a = -1 and b = -6. 3. Find the exponential of the matrix of the system by using the diagonalization approach.

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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Problem 3: Consider the IVP y" + ay' + by = e³*, y(0) = 1, y'(0) = 1. 1. Write the second-order IVP as an IVP of a first-order system of equations by setting y' = z. Show that the eigenvalue of the resulting matrix are the same as the roots of the characteristic polynomial that is associated to the second-order IVP. 2. For the remainder of this question, set a = -1 and b = -6. 3. Find the exponential of the matrix of the system by using the diagonalization approach. 4. Use the variation of parameters formula together with the exponential from part 3) to find the solution of the IVP in systems form. 5. Solve the second-order IVP and compare.
Transcribed Image Text:Problem 3: Consider the IVP y" + ay' + by = e³*, y(0) = 1, y'(0) = 1. 1. Write the second-order IVP as an IVP of a first-order system of equations by setting y' = z. Show that the eigenvalue of the resulting matrix are the same as the roots of the characteristic polynomial that is associated to the second-order IVP. 2. For the remainder of this question, set a = -1 and b = -6. 3. Find the exponential of the matrix of the system by using the diagonalization approach. 4. Use the variation of parameters formula together with the exponential from part 3) to find the solution of the IVP in systems form. 5. Solve the second-order IVP and compare.
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