#$&* course MTH 279 Query 11 Differential Equations*********************************************
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: 3.8.6. Euler's method applied to the equation y ' = alpha t + beta, y(t_0) = y_0, yields y values -1, -.9, -.81 and -.73 at respective t values 0, .1, .2, .3. Find the values of alpha, beta, t_0 and y_0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Assuming that t_ 0 = 0, and y_0 = -1 (the corresponding pair). y_1 = y_0 + 0.1(αt+β) = -1+ 0.1β y_2 = -1 + 0.1β + 0.1(0.1α + β) = -0.9 y_3 = -0.9 + 0.1(0.2α+β) = -0.81 y_4 = -0.81 + 0.1(0.3α+β) = -0.73 β = 0.8-0.3α -0.9+0.1(0.2α+0.8-0.3α) = -0.81 α = -1 β = 1.1 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: 3.8.8a. For each of the following situations, will Euler's method overestimate or underestimate the values of the solution to an equation: • The solution curve is known to be increasing and concave up. • The solution curve is known to be increasing and concave down. • The solution curve is known to be decreasing and concave up. • The solution curve is known to be decreasing and concave down. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If the solution curve is known to be increasing and concave up, Euler’s method will underestimate. If the solution curve is increasing and concave down, Euler’s method will overestimate. If the solution curve is decreasing, and concave up, Euler’s method will underestimate. If the solution curve is decreasing and concave down, Euler’s method will overestimate. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: 3.8.14. y ' = y^2 with y(0) = 1. Solve the equation. Perform Euler's Method to approximate the values of the solution on the t interval [0, 1.2] with step size h = .1. Compare the values you get with the values given by your solution to the equation. This could be done by hand, but it would take awhile and the probability of an error would be relatively high. A spreadsheet is recommended. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Euler’s Method Results: y(0) = 1 y(0.1) = 1.1 y(0.2) = 1.22 y(0.3) = 1.37 y(0.4) = 1.56 y(0.5) = 1.80 y(0.6) = 2.12 y(0.7) = 2.58 y(0.8) = 3.24 y(0.9) = 4.29 y(1) = 6.13 y(1.1) = 9.89 y(1.2) = 19.66 Actual Values: y(0) = 1 y(0.1) = 1.11 y(0.2) = 1.25 y(0.3) = 1.43 y(0.4) = 1.67 y(0.5) = 2 y(0.6) = 2.5 y(0.7) = 3.33 y(0.8) = 5 y(0.9) = 10 y(1) = undef y(1.1) = -10 y(1.2) = -5 The estimates by Euler’s method are pretty close, although not always accurate, especially in the region of the vertical asymptote. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK"