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Letting y₁ (t) = 0(t) and y2(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving yı and Y2. Summarise the system in the matrix equation y' = My. Show that the eigenvalues of M are identical to the roots computed in (i)(b). Find the eigenvector for the case of repeated eigenvalues. For this matrix you be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found. should only

Letting y₁ (t) = 0(t) and y2(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving yı and Y2. Summarise the system in the matrix equation y' = My. Show that the eigenvalues of M are identical to the roots computed in (i)(b). Find the eigenvector for the case of repeated eigenvalues. For this matrix you be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found. should only

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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Please solve for part ii; a, b and c. The a, b, c answers for part i are attached

@ The Characteristic ea" is 1² + 1/ 1 + 1/² = 0 6 Condition: - 6²=41K 710 (where 16,470) @ Ⓒ General Sol" for repeated real roots; ol) = (a +(₂+) é (1/25) + where cy cz are t t) arbitrary Constants.
Transcribed Image Text:@ The Characteristic ea" is 1² + 1/ 1 + 1/² = 0 6 Condition: - 6²=41K 710 (where 16,470) @ Ⓒ General Sol" for repeated real roots; ol) = (a +(₂+) é (1/25) + where cy cz are t t) arbitrary Constants.
The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + I dt I dt² (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door (i) (a) Substitute ( Ө + = 0 wall e(t) = = et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). (ii) (a) Letting y₁(t) = : 0(t) and y₂(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving y₁ and Y2. Summarise the system in the matrix equation y' = M y. (b) Show that the eigenvalues of M are identical to the roots computed in (i)(b). (c) Find the eigenvector for the case of repeated eigenvalues. For this matrix you should only be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found.
Transcribed Image Text:The behaviour of a swinging door can be modelled by the second order, constant coefficient ODE d²0 b de k + I dt I dt² (1) where (t) gives the angular position of the door, and I, b, k are positive mechanical parameters that affect the motion of the door. wall door (i) (a) Substitute ( Ө + = 0 wall e(t) = = et into equation (1) to find the characteristic equation. (b) Determine the relationship between I, b, and k that will ensure the roots of the characteristic equation are not complex. That is, we seek solutions that have no oscillatory behaviour. (c) State the general solution corresponding to the case of repeated real roots in (b). (ii) (a) Letting y₁(t) = : 0(t) and y₂(t) = 0'(t) = y₁ (t), show that equation (1) can be written as a system of first order ODEs involving y₁ and Y2. Summarise the system in the matrix equation y' = M y. (b) Show that the eigenvalues of M are identical to the roots computed in (i)(b). (c) Find the eigenvector for the case of repeated eigenvalues. For this matrix you should only be able to find a single eigenvector. Repeated eigenvalues having a single eigenvector are referred to as defective eigenvalues. The matrix is also said to be defective. They will be explored in other units. It is common to revert to the procedure of (i)(a)-(c) when a defective eigenvalue is found.
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