Query 13 Differential Equations

Question:  Find the largest interval on which the equation

y '' + y ' + 3 t y = tan(t)

has a solution, with the initial conditions y(pi) = 1 and y ' (pi) = -1.

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Question:  Find the largest interval on which the equation

t y '' + sin(2 t) / (t^2 - 9) y ' + 2 y = 0

has a solution, with the initial conditions y(1) = 0, y ' (1) = 1.

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Question:  Tell whether each of the following is increasing or decreasing, and whether concave down or concave up, in the vicinity of the initial point:

• y '' + y = 2 - sin(t), y(0) = 1, y ' (0) = -1.

• y '' + y = - 2 t, y(0) = 1, y ' (0) = -1.

• y '' - y = t^2, y(0) = 1, y ' (0) = 1.

• y '' - y = - 2 cos(t), y(0) = 1, y ' (0) = 1.

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