query 13

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course Phy 279

3/6 12

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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Your solution:

t can’t be (`pi/2 + n`pi), where n is any integer or zero

Interval is then 2`pi

Note no restrictions on y

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:

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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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Your solution:

t cannot equal 3 or -3

Interval is then -3

No restrictions on y

confidence rating #$&*:

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Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

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Self-critique rating:

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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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Your solution:

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

o y’’(0) = 1 and y’(0) = -1

o so it is decreasing at a increasing rate, concave down

@& y '' ( 0 ) = 1 makes this concave up at the pont, so the function is decreasing and concave up, i.e., decreasing at a decreasing rate.

*@

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

o y’’(0) = -1 and y’(0) = -1

o so it is decreasing at a decreasing rate, concave up

@& This one would be decreasing and concave down, therefore decreasing at an increasing rate.*@

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

o y’’(0) = 1 and y’(0) = 1

o so it is increasing at a increasing rate, concave up

@& good*@

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

o y’’(0) = -1 and y’(0) = 1

o so it is increasing at a decreasing rate, concave down

@& good*@

&#Good responses. See my notes and let me know if you have questions. &#

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