Question

2. Solve the following initial value problems using the fact that the differential equations below are separable: a. tºy' = (t + 1)y, y(1) = 2, t > 0 b. y' = –2t tan(y), y(0) = 5

Answer 1

27a) t ² g = (2+1 y (1) = 2, A TO on, die dy - (A+1) at 42 Integrating weget, (& 1 at dy af s dat adt t 49 1 t =enti +] nel t2 or, enlyn St i en lyl- en/tl-. [ Senda te (@z Arbitrary const) Aron, y (1)=2 * en 2 = ln 1-1te te = 1 + en 2 [..ln1 ao] Hence from we get en lyl=en 1tl - 1 ta tilna on, enlol = ene1 + (1 - 4 J [en1241 +(+-) J = e en 1241 e(t-14) :. y e(t - # [etna ex e at e = 27

= -2f dt by yl=-2t tany y (0) = 1/2 on, dy - 27 dt tany OM, cosy dy sing Integrating, ſeose dy = -22d4 sing x ² te on, en/singl -8000) = 1/2 + ensino Hence froomo of e=Arbitrary Const.) T-Siwy, 1 te= en en/singl = -t² 2 on sing ent2 Sini (et' (ans) a