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course MTH 279
3.1For 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
y’ = sqrt(t)/cos(y)
when y is pi/2 + n*pi the function is undefined, thus if y(pi) = 1 we know the parameters lie as -pi/2
2. (t^2 - 1) + (1 + y) y ' = 0, y(0) = 1
y’ = -(t^2 - 1) / (1+y)
if y = -1 the function is undefined
when y is 0 the function is 1 , thus when y = 0 t= 0
-infinity < y< -1 U -1< y < infinity -infinity < t < infinity
3. cos(t) y ' = 1 - tan(y), y(pi/4) = 0
y’ = (1-tan(y)) / cos(t)
when t = pi/2 + n*pi the function is undefined
thus -pi/2 < t< pi/2
-infinity < y < infinity
Write each equation in Problems 4-6 in the form y ' = f(t, y), calculate the y partial derivative f_y(t, y) and determine the regions in the plane in which both f and f_y are continuous.
4. t y ' + e^y = 0
y’ = -(e^y )/t
f(t,y) = -infinity < t < 0 and y is all real numbers
fy(t,y) = -(e^y)/t which is the same parameters thus the domain is the same
5. sin(t) y ' + sqrt(y) = 0
y’ = -sqrt(y) / sin(t)
f(t,y) = 0< t < pi and 0 <= y < infinity
fy(t,y) = -1/2 (y)^-1/2 * (1/sin(t)) , thus the new domain is the same for t, but for y, 0 < y < infinity, thus the total domain is from 0 exclusive to pi exclusive.
6. cot(t) y’ + 1 / (y^2 - 1) = 0
y’ = (1/((y^2-1)(cot(t))))
f(t,y) = -infinity < y < -1 U -1< y < 1 U 1 < y < infinity and -pi/2< t < pi/2
fy(t,y) = -2y* tan(t) / (y^2 -1)^2 y and t still have the same domains thus the total domain is -pi/2 < y’ < -1 U -1< y’ < 1 U 1 < y’ < pi/2
7. For the equation of #4, if we are given the initial condition y(1) = 2, what is the largest region of the plane which contains the corresponding point?
-infinity < y’ < 0 U 0 < y’ < infinity
8. For the equation of #5, if we are given the initial condition y(-1) = 2, what is the largest region of the plane which contains the corresponding point?
(2 sin(-1))^2 < y < infinity
9. For the equation of #6, if we are given the initial condition y(0) = 1, what is the largest region of the plane which contains the corresponding point?
Y is exactly 0
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