For the given initial value problem, Rewrite Compute differential equation, if n

Question

For the given initial value problem, Rewrite Compute differential equation, if necessary, to obtain the form y' = f(t, y), y(t0) = y0. Identify the function f(t, y). Compute f/ y. Determine where in the ty-plane both f(t, y) and f/ y are continuous. Determine the largest open rectangle in the ty-plane that contains the point(t0, y0) and in which the hypotheses of Theorem 2.2 are satisfied. 3y' + 2t cos y = 1, y(pi/2) = -1 3t' + 2 cos y = 1, y(pi/2) = -1 2t + (1 + y2)y' = 0, y(1) = 1 2t + (1 + y3)y' = 0, y(1) = 1 y' + ty1/3 = tant, y(-1) = 1 (y2 - o)y' + e-y = t2, y(2) = 2 (cos y)y' = 2 + tan t, y(0) = 0 (cos 2t)y' = 2 + tan y, y(pi) = 0

Explanation / Answer

5)y`=tant-ty1/3

f(t,y)=tant-ty1/3 continuous

f/y=ty-2/3/3 discontinuous at y=0

6)(y2-9)y`=t2-e-y

f(t,y)=t2-e-y/(y2-9)continuous

f/y=e-y{y(2-2t2ey)+y2-9}/(y2-9)2 discontinuous at t=0

5)cos2ty`=2+tany

f(t,y)=(2+tany)/cos2t discontinuous at many points

f/y=sec2y*sec2t discontinuous at many points

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