Problems In each of Problems 1 through 4, determine (without solving the how solutions appear to behave as t increases and how their behavio problem) an interval in which the solution of the given initial value depends on the initial value yo when t = 0. problem is certain to exist. 1. (t - 5) y' + (Int)y = 2t, y (1)= 2 13. Gy'= ty (4 -y) 14. Gy'= y (4 -ty) 2. y' + (tant) y = sint, y (27) = 0 Isnolletugnoo to 20m 60/15. Gy'=t-1-y In blood? 16. G y' = 1 -2 -y? To rigung oth 9isone of ban a Ynso of s 3. (16 - 12) y'+ 2ty = 312, y (1)=-5 17. Consider the initial value problem y' = y!/3, y(0) = 0 from 4. (In (t)) y' + 2y = cot (t), y(2) =3 101 evloz of vans al li doirlw mi an Example 2.4.3 in the text. bas bus a. Is there a solution that passes through the point (1, 1)? If sc In each of Problems 5 through 8, state where in the ty-plane the find it. hypotheses of Theorem are satisfied. joWlal orIT .oldshiv inobriorobot b. Is there a solution that passes through the point (2, 1)? If so 5. y' = (4 - 12 - 2)1/2 6 . dy _ (cott) y asd 10 y ni pizzabang find it. diebol+ yivlang yd yoko c. Consider all possible solutions of the given initial value prob 7. y' = (12 + 1/2)5/2 4+12 8. y' = 2y - yz 2891929 lipilami me bad lem. Determine the set of values that these solutions have a 1 = 2. non 19bTO-1 In each of Problems 9 through 12, solve the given initial value 18. a. Verify that both y, (1) = 1 - t and y,(1) = -1/4 are solutions problem and determine how the interval in which the solution exists of the initial value problem 100 Isbiomull 10 lapin depends on the initial value yo- 9. y' = -2t/y, y (0) =yo y (2) = -1. 2 10. y' = 4ty, y (0) = yo 12 Where are these solutions valid? . Isunionibn 11. y' = V(1+13)' y (0) = yo arly lo anomen gumvilmabe at tub. Explain why the existence of two solutions of the given 13 Lcollins boliweb sjoin them word problem does not contradict the uniqueness part of Theorem 12. y' = y (1+ 14 )' y (0) = Yo ots enoilsups longgollob 79610-121 2.4.2. gnom c. Show that y = ct + c', where c is an arbitrary constant, In each of Problems 13 through 16, draw a direction field and plot (or To goiz satisfies the differential equation in part a for ( 2 - 2c. sketch) several solutions of the given differential equation. Describe dhoomun sdid