#$&* course MTH 279 4/15 4 Query 25 Differential Equations*********************************************
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I do not know if there was another test for linear independence I wasn't considering. ------------------------------------------------ Self-critique rating: 3 ********************************************* 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] YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Yes. det = 2-2cos^2t, which is nonzero on -inf < t < inf
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* 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 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: P = [ 2/t , 2/t^2 ; -2 , 0]
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y_1 = [e^t ; e^t] y_2 = [e^-t ; -e^-t] [e^t ; e^t] = [0, 1; 1, 0] * [e^t ; e^t] [-e^-t ; e^-t] = [0, 1; 1, 0] * [e^-t ; -e^-t] doing matrix multiplication on the right hand sides of these two equations, we do see [e^t ; e^t]= [e^t ; e^t] and [-e^-t ; e^-t] = [-e^-t ; e^-t] psi = [ e^t , e^-t ; e^t , -e^-t] det(psi) = -2, so yes. y_1 and y_2 are a fundamental set. [ e^t , -e^-t ; e^t , e^-t] = [0, 1; 1, 0]*[ e^t , e^-t ; e^t , -e^-t] yes, psi is a solution. C = [-4 , -2 ; 2 , -6] since C is invertible, psi_hat is a solution matrix. it is also a fundamental matrix. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: psi is not a fundamental matrix. [ e^t, -2e^(-2 t); 0, -2e^(-2 t) ] is not equal to [1 , 1 ; 0 , -2]*[ e^t, e^(-2 t); 0, e^(-2 t) ] = [e^t , 2e^(-2t) ; 0 , -2e^(-2t)] confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I found it odd that the right hand side differed from the left by a negative sign on only the 1,2 term, but I could not find where I made an error in my calculations. ------------------------------------------------ Self-critique rating:3