Query 11 Differential Equations

Question:  3.8.4.  Solve the equation y ' = - y + t with y(0) = 0.

Write the expression y_(k + 1) = y_k + h f (t_k, y_k) for h = .01.

Find y_k for k = 0, 1, 2, 3.

Using your original solution for the equation, compare your values of y_k with the values given by the accurate solution.

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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.

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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.

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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.

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