Query 25

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course MTH 279

Query 25 Differential Equations*********************************************

Question: Determine whether the set of solutions {y_1, y_2, y_3} is linearly independent, where

y_1 = [ e^t, 1]

y_2 = [ e^(-t), 1]

y_3 = [ sinh(t), 0]

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Your solution:

Ay_1+By_2+Cy_3 = 0

[Ae^t, A; Be^(-t), B; Csinh(t), 0] = [0;0;0]

Using row reduction on

[Ae^t, A, 0; Be^(-t), B, 0; Csinh(t), 0, 0]

yields [1,0,0; 0,1,0; 0,0,0]. This means that the third row (y_3) is a linear combination of the first and second rows. This makes sense since sinh(t) = e^t/2 - e^(-t)/2, which is a linear combination of the first and second solution.

We can determine that the set is not linearly independent.

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Good.

You could also have used the Wronskian, which would be zero.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Determine whether the set of solutions {y_1, y_2, y_3} is linearly independent, where

y_1 = [ 1, sin^2(t), 0]

y_2 = [ 0, 2 - 2 cos^2(t), -2]

y_3 = [ 1, 0, 1]

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Your solution:

Ay_1 + By_2 + Cy_3 = 0

Using row reduction on our y:

[1, sin^2(t), 0, 0; 0, 2-2cos^2(t), -2, 0; 1, 0, 1, 0] we get:

[1,0,1,0; 0,1, -1/sin^2(t), 0; 0, 0, 0, 0]

This means that the system is linearly dependent. That is, the third solution is a linear combination of the first two.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Determine whether there is a matrix P(t) such that

y_1 = [ t^2, 0 ]

y_2 = [ 2t, 1 ]

is a fundamental set of solutions to the equation

y ' = P(t) y.

If so, find such a matrix P(t).

Hint: The matrix psi(t) = [y_1, y_2 ] = [ t^2, 2 t; 0, 1 ] would need to satisfy psi ' (t) = P(t) psi(t).

In standard notation we could write this as follows:

satisfies

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Your solution:

Ψ = [t^2,2t; 0,1]

Ψ’ = [2t, 2; 0, 0]

Ψ’ = A*Ψ

A = Ψ’*Ψ^(-1)

Ψ^(-1) = 1/det(Ψ)*[1, -2t; 0, t^2] = 1/(t^2)* [1, -2t; 0, t^2] = [1/t^2, -2/t; 0,1]

A = [2t, 2; 0, 0]*[1/t^2, -2/t; 0,1] = [2/t, -2; 0, 0]

We can plug this back in and verify that it is true.

confidence rating #$&*:

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Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: If the matrix psi(t) = [y_1, y_2] = [e^t, e^(-t); e^t, - e^(-t)]:

What are the vector functions y_1 and y_2?

Write out the system of two differential equations represented by the equation y ' = P(t) y with P(t) = [0, 1; 1, 0].

Show that y_1 and y_2 are both solutions of the equation y ' = P(t) y with P(t) = [0, 1; 1, 0].

Show that { y_1 , y_2} is a fundamental set for this equation.

Show that the matrix psi(t) is a solution of the matrix equation psi ' = P(t) psi.

Show that the matrix psi(t) is a fundamental matrix for the linear system of equations.

Let psi_hat(t) = [ 2 e^t - e^(-t), e^t + 3 e^(-t); 2 e^t + e^(-t), e^t - 3e^(-t) ].

Find a constant matrix C such that psi_hat(t) = psi(t) * C.

Based on your matrix C, is psi_hat(t) a solution matrix for the system?

Based on your matrix C, is psi_hat(t) a fundamental matrix for the system?

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Your solution:

y_1 = [e^t; e^t]

y_2 = [e^(-t); -e^(-t)]

y_1’ = y_2

y_2’ = y_1

y_1’ = [0,1;1,0][e^(-t); -e^(-t)] = [-e^(-t); e^(-t)]

y_1’ = [-e^(-t); e^(-t)] = y_2

Substitution shows that both are satisfied.

W(t) = det([e^(t), e^(-t); e^(t), -e^(-t)]) = -2 ≠ 0 for all values of t, so y_1 and y_2 form a fundamental set.

Ψ’ = [e^(t), -e^(-t); e^(t), e^(-t)] = [0,1;1,0]*[e^(t), e^(-t); e^(t), -e^(-t)] = [e^(t), -e^(-t); e^(t), e^(-t)]

W(t) = det(Ψ) = -2 ≠ 0 for all values of t, so Ψ is a fundamental matrix.

C = Ψ^(-1)*Ψ_hat

C = [2,1;-1,3]

Based on this, Ψ_hat is a solution matrix to the system, and is also a fundamental matrix, because it is a fundamental matrix (Ψ) multiplied by a constant matrix.

@&

To draw this conclusion C would have to be invertible. Its determinant is 7, which is nonzero, so that is the case. Thus psi or psi_hat is a fundamental set, so is the other.

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confidence rating #$&*:

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Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Given the system

y ' = [ 1, 1; 0, -2 ] y

verify that

psi(t) = [ e^t, e^(-2 t); 0, e^(-2 t) ]

is a fundamental matrix for the system.

Find a matrix C such that

psi_hat(t) = psi(t) * C

is a solution matrix satisfying initial condition psi_hat(0) = I, where I is the identity matrix.

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Your solution:

W(t) = det(Ψ) = e^(-t) ≠ 0 for all values of t, so it is indeed a fundamental matrix of the system.

Ψ_hat = Ψ*[1,-1;0,1] = [e^(t), e^(2t)-e^(t); 0, e^(-2t)]

Ψ_hat(0) = [1,0; 0,1] = I

confidence rating #$&*:

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Given Solution:

&#Good responses. See my notes and let me know if you have questions. &#