#$&* course Mth 279 7/21 10:44 pm Query 15 Differential Equations*********************************************
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Are y1 = 2 e^(-2 t) cos(t) and y2 = e^(-2 t) sin(t) solutions to the equation y '' + 4 y ' + 5 y = 0? What are the initial conditions at t = 0? Is {y1, y2} a fundamental set? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y1' = -4e^(-2t)cos(t) + 2e^(-2t)sin(t) y1'' = -8e^(-2t) cos(t) + 4e^(-2t) - 4e^(-2t)sin(t) +2e^(-2t) cos(t) Plugging in: [-8e^(-2t) cos(t) + 4e^(-2t) - 4e^(-2t)sin(t) +2e^(-2t) cos(t) ] + 4[-4e^(-2t)cos(t) + 2e^(-2t)sin(t)] + 5[e^(-2t)cos(t)] = 0 e^(-2t)[-8cos(t) + 2cos(t) -16cos(t) + 10 cos(t) +4sin(t) - 4sin(t) + 8sin(t) ] = 0 e^(-2t)(-12cos(t) + 8sin(t) ) does not equal 0 y1 is not a solution to the equation. y2' = -2e^(-2t)sin(t) + e^(-2t) cos(t) y2'' = 4e^(-2t) sin(t) - 2e^(-2t)cos(t) - 2e^(-2t)cos(t) - e^(-2t)sin(t) plugging in: [4e^(-2t) sin(t) - 2e^(-2t)cos(t) - 2e^(-2t)cos(t) - e^(-2t)sin(t)] + 4[-2e^(-2t)sin(t) + e^(-2t) cos(t)] + 5[e^(-2t)sin(t)] = 0 e^(-2t)[4sin(t) - sin(t) - 8 sin(t) + 5sin(t) - 2cos(t) - 2cos(t) + 4 cos(t) ] = 0 e^(-2t) [0] = 0 0=0 y2 is a solution. I don't think this is right, but I could not find my error in testing y1 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: y1_bar = 2 y1 - 2 y2 and y2_bar = y1 - y2. Is {y1_bar, y2_bar} a fundamental set? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: if y1 = 2y2 W = [y1 2y1, y1' 2y1'] 2*y1*y1' - 2*y1*y1' = 0 Yes, it is a fundamental set confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Note that y_1_bar = 2 * y_2_bar. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Is {e^t, 2 e^(-t), sinh (t) } a fundamental set on the interval (-infinity, infinity)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: yes because none of the solutions have asymptotes. What is the difference between sinh and sin???
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: