gIVE DETAILED SOLUTION FOR EACH OF THE OPTIONS Given a diff

** gIVE DETAILED SOLUTION FOR EACH OF THE OPTIONS

Given a differential equation dy/dt = f(t,y), y(t_0) = y_0, a unique solution is at guaranteed to exist in a rectangle which contains the point (t_0,y_0) in the ty-plane where f(t,y) and f(t,y) and f/y (t, y) are both continuous. Which of the following differential equations have a unique solution near the initial value? I. dy/dt + ty = t + 1/t, y (1) = 0 II. dy/dt + ty = t + 1/t, y (0) = 1 III. dt/dt = y^1/3, y(0) = 0 A. I only B. II only C. III onlyh D. I and II E. II and III

Solution

(I) The function f(t,y)= t +1/t -ty and f1(t,y) are continous on (1,infiinity)

I has unique solution

(II) The function f(t,y)= t +1/t -ty is not continous at t=0

II has no unique solution

(IIi) The function f(t,y)= t +1/t -ty is continous but not derivable at t=0

II has no unique solution