Question

makes) For each of the following direction field plots, write down a function f(ty) such that! the differential equation dy = f(t,y) could have this direction field. In each case, give reasons why you think your differential equation could have the direction field shown. + + + to TTTTTTT 1 1 1 1 - 111111111111 PTT TTT 1 1 1 1 1 1 1 1 1 1 1 1 1 UTILIITTI 1 1 1 1 1 1 1 1 III -1 1 1 1 1 1 | | -0.51 I | II III + ! ! 1 1 1 1 1 1 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -3111111111 IIIIIIIIIIIIIII 1 1 1 1 1 1 1 ! ! -15 ! ! ! ! - T2 TT 0 1 4 5 7 9 10 1 1 1 1 1 1 1 1111111111 1 1 1 1 1 III III III 1 1 1 1 1 1 LLL -

Answer 1

An ordinary differential equation tells us about the tangent of the trajectory at any point (t,y) in the t-y plane. So direction field indicates the rate of change of the trajectory starting from any initial point.

See in detail:

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Thu he -that, yt-25, y 2.5 (t-)e 2.5 y(t)=o 04 y y 7 blt y-, i Shou Tuit he >y 2.5 Should be ( Svadradie the forh Ty fonota yy2.5)

il, he fee that di keitin For the 2nd thftow Coomaomy 7he ny ditfenent enatn

Forthe 3ra veetin el, he fee that, forto,o 2 7h fom -2) Thug y y tu e dtterent e y tte frm dy t6- dt