Consider the solutions y 1 (t) = [e -3t , e -3t , 0] T and y 2 (t) = [0, e -5t , e -5t ] T to the homogeneous linear system of differential equations y'(t) = Ay(t), where A is the 3 × 3 matrix: A = [-5 2 -2] [ 1 -4 -1] [-1 1 -6] Which of the following statements is true?

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Mathematics · Differential Equations

Consider the solutions y 1 (t) = [e -3t , e -3t , 0] T and y 2 (t) = [0, e -5t , e -5t ] T to the homogeneous linear system of differential equations y'(t) = Ay(t), where A is the 3 × 3 matrix: A = [-5 2 -2] [ 1 -4 -1] [-1 1 -6] Which of the following statements is true?

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y 1 and y 2 are linearly independent but do not form a basis for the set of all solutions to the system.

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Answer 1

y 1 and y 2 are linearly dependent.

Answer 2

There exists another solution y 3 such that {y 1 , y 2 , y 3 } forms a basis for the set of all solutions to the system.

Answer 3

y 1 and y 2 are linearly independent but do not form a basis for the set of all solutions to the system.

Answer 4

y 1 and y 2 form a basis for the set of all solutions to the system.

Consider the solutions y 1 (t) = [e -3t , e -3t , 0] T and y 2 (t) = [0, e -5t , e -5t ] T to the homogeneous linear system of differential equations y'(t) = Ay(t), where A is the 3 × 3 matrix: A = [-5 2 -2] [ 1 -4 -1] [-1 1 -6] Which of the following statements is true?

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Consider the solutions y 1 (t) = [e -3t , e -3t , 0] T and y 2 (t) = [0, e -5t , e -5t ] T to the homogeneous linear system of differential equations y'(t) = Ay(t), where A is the 3 &times; 3 matrix: A = [-5 2 -2] [ 1 -4 -1] [-1 1 -6] Which of the following statements is true?

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Consider the solutions y 1 (t) = [e -3t , e -3t , 0] T and y 2 (t) = [0, e -5t , e -5t ] T to the homogeneous linear system of differential equations y'(t) = Ay(t), where A is the 3 × 3 matrix: A = [-5 2 -2] [ 1 -4 -1] [-1 1 -6] Which of the following statements is true?