Question

4. Find the longest r-interval where the initial value problem: y'+ty: = tany, y(-1) = 1 has a unique solution. (10 points)

Answer 1

There is no x in the given question,it contains t and y.So the question should find the longest t-internal in place of find the longest x- interval.

Theorem are on Existence of a Unique solution for a first order DE: dy =tex,yy, yeaol=yo dx If ferrys and of an continuous rectangular defined by a calb, clyld contains the point (2o, Yo), an internal then and defined unique function yea) the infernal that satisfien the IND region defined that there existi centered at a ro

Giren ulog o'tty' Linot o y (-12-1 स 13 -ty @ =) y'a tany Comparing it with y' Fitiy ICH - tony 113 ty t of t = secry te he 1/3 3 in dine =0 ) and of af 3y where Elhy confirmour cosy nez =) Y-2-4) in din continuous ту) where cosy=0 y=0 =) y=0, O, (2nH) notice that both Fit, y) and 2 ore simultaneo continuous for x < t ca or we Gunoa

is geven an Intitial condition y (11=1 and -1€ (-0, d) Long est t-interval where IVP has unique solution is ,^, (real