#$&* course MTH-279 07/29 around 9:55PMJust turning in some homework completed leading up to test two. Query 14 Differential Equations
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Decide whether y_1 = e^(-t) and y_2 = 2 e^(1 - t) are solutions to the equation y '' + 2 y ' + y = 0. If so determine whether the two solutions are linearly independent. If the solutions are linearly independent then find the general solution, as well as a particular solution for which y (0) = 1 and y ' (0) = 0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y1 = e^-t y1’ = -e^-t y1’’ = e^-t y2 = e^(1-t) y2’ = -e^(1-t) y2’’ = e^(1-t) e^-t - 2e^-t + e^-t = 0 e^(1-t) - 2e^(1-t) + e^(1-t) = 0 Using the Wronskian method: (e^-t * -e^1-t) - (e^(1-t) * -e^-t) = 0 Linearly dependent and not a fundamental set. Confidence rating: 3
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Given Solution: Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Suppose y_1 and y_2 are solutions to the equation y '' + alpha y ' + beta y = 0 and that y_1 = e^(2 t). Suppose also that the Wronskian is e^(-t). What are the values of alpha and beta? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y1 = e^2t y1’ = 2e^2t y1’’ = 4e^2t 4e^2t + 2alpha e^2t + beta e^2t = 0 4 + 2alpha + beta - eq1 f(t)g’(t) - f’(t)g(t) y1y2’ - y1’y2 = e^-t e^2t y2’ - 2e^2t y2 = e^-t (y2 e^2t)’ = e^-t integral (y2 e^2t)’ = integral e^-t y2 e^2t = -e^-t y2 = -e^-3t ---------- y2 = -e^-3t y2’ = 3e^-3t y2’’ = -9e^-3t -9e^-3t + 3alpha e^-3t - beta e^-3t = 0 -9 + 3alpha - beta = 0 -eq 2 ----- 4 + 2alpha + beta = 0 -9 + 3alpha - beta = 0 ----- -5 + 5 alpha = 0 5alpha = 5 alpha = 1 ----- 4 + 2 + beta = 0 beta = 6 Confidence rating: 3
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Suppose y_1 and y_2 are solutions to the equation y '' + alpha y ' + beta y = 0 and that y_1 = e^(2 t). Suppose also that the Wronskian is e^(-t). What are the values of alpha and beta? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y1 = e^2t y1’ = 2e^2t y1’’ = 4e^2t 4e^2t + 2alpha e^2t + beta e^2t = 0 4 + 2alpha + beta - eq1 f(t)g’(t) - f’(t)g(t) y1y2’ - y1’y2 = e^-t e^2t y2’ - 2e^2t y2 = e^-t (y2 e^2t)’ = e^-t integral (y2 e^2t)’ = integral e^-t y2 e^2t = -e^-t y2 = -e^-3t ---------- y2 = -e^-3t y2’ = 3e^-3t y2’’ = -9e^-3t -9e^-3t + 3alpha e^-3t - beta e^-3t = 0 -9 + 3alpha - beta = 0 -eq 2 ----- 4 + 2alpha + beta = 0 -9 + 3alpha - beta = 0 ----- -5 + 5 alpha = 0 5alpha = 5 alpha = 1 ----- 4 + 2 + beta = 0 beta = 6 Confidence rating: 3
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&! ********************************************* Question: Suppose y_1 and y_2 are solutions to the equation y '' + alpha y ' + beta y = 0 and that y_1 = e^(2 t). Suppose also that the Wronskian is e^(-t). What are the values of alpha and beta? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y1 = e^2t y1’ = 2e^2t y1’’ = 4e^2t 4e^2t + 2alpha e^2t + beta e^2t = 0 4 + 2alpha + beta - eq1 f(t)g’(t) - f’(t)g(t) y1y2’ - y1’y2 = e^-t e^2t y2’ - 2e^2t y2 = e^-t (y2 e^2t)’ = e^-t integral (y2 e^2t)’ = integral e^-t y2 e^2t = -e^-t y2 = -e^-3t ---------- y2 = -e^-3t y2’ = 3e^-3t y2’’ = -9e^-3t -9e^-3t + 3alpha e^-3t - beta e^-3t = 0 -9 + 3alpha - beta = 0 -eq 2 ----- 4 + 2alpha + beta = 0 -9 + 3alpha - beta = 0 ----- -5 + 5 alpha = 0 5alpha = 5 alpha = 1 ----- 4 + 2 + beta = 0 beta = 6 Confidence rating: 3
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!#*&!