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Mth 279
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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3.1
For each of the following equations, each of the form y ' = f(t, y), find the largest open set of the form a < t < b, alpha < y < beta which contains the point (t_0, y(t_0)) implied by the given initial condition, and for which the functions f and f_y are both continuous.
1. y ' cos(y) - sqrt(t) = 0, y(pi) = 1
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I'm working on the problem set for chapter 3 and I just can't figure out what these directions are asking me to find. I know that if I solve for y' its: y’ = (sqrt (t)) / (cos(y)). I also know that on page 61 in my book, is theorem 3.1, which im thinking i would use, but don't know how to start these problems. Can you walk me through this one problem and I'm sure I can do the rest. The book really doesn't give a clear explanation of this process.
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@& sqrt(t) / cos(y) is undefined where cos(y) = 0, and also where t < 0.
cos(y) = 0 when y = pi / 2 + n * pi. The lines y = pi/2 + n pi are horizontal lines separated by distance pi. The two lines closest to the x axis are y = -pi/2 and y = pi / 2.
t >= 0 includes the entire half-plane from the y axis to the right, including the y axis.
y(pi) = 1 means that y = 1 when t = pi. The point (pi, 1) lines between y = -pi/2 and y = pi/2, so the region of the plane which contains this point is
0 <= t < infinity
-pi/2 < y < pi/2
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