Advanced Calculus fi..

Chapter 9 Ordinary Differential Equations 6572. a) Show that F(t, x) = tlxJ is continuous for all (t, x), that F, exists and is continuousexcept for x = 0, and that F satisfies local Lipschitz conditions for all (t, x) (as in(9.61) with n = 1). ib) Find all solutions of the differential equation dxldt = tlxl and show that a uniquesolution passes through each point.3. a) Show that F(t, x) = x4/' is continuous and has a continuous derivative F, for all(t, x).b) Find all solutions of the differential equation dxldt = x4I3 and show that there is aunique solution through each point.i4. a) Show that F(t, x) = x1I3 is continuous for all (r, x) but that F, does not exist forx = 0 and that F(t, x) fails to satisfy a local Lipschitz condition for all (t, x).b) Obtain all solutions of the differential equation dx/dt = x113 and verify that there ismore than one solution through each point on the t-axis.5. For each of the following differential equations, find x = +(t; 7, x), the complete solutionthrough (7, f), and determine whether x = +(t; 0, f ), gives all solutions:a) - 2x = e', b) 2 = t sinx,C) = e'x2, d) 2 + ,*yx2 = 0 (x > 0).d re) + = 0, all (t. x) except (0,O).6. Let the following system, with initial condition, be given:dx- =tx-y, 9=tx2-y, x=l and y=l fort=O.dtdta) Show that the solution can be obtained for It 1 < 0.3 by successive approximations.b) Ohtain the first three successive approximations for x and y.c) Obtain the power series solution through terms in t3 and compare with the result ofpart (b).7. Obtain the complete solution satisfying the given initial conditions:a) y" = evy', y = 0 and y' = 1 for x = 0;b) dx/dt=ycost+t+x, dy/dt= 1+y2, x=O, y =Ofort=O;c) dxldt = 4x - 5y, dy/dt = x - 2y, x = xo, y = yo fort = to.8. a) Let x = $(t), It - 71 p h, $(i) = 2, where $'(t) is continuous and it is known thatI$'(t)( 5 M whenever (t, $(r)) is in the rectangle R: It - il p h, Ix - f 1 5 k, whereMh 5 k. Show that (1, $(t)) is in R for It -TI 5 h.b) Generalize the result of part (a): Let x, = +, (t), It - t 1 ( h, $,ti) = f, (i =1, ..., n), where +,'(t) is continuous and it is known that I$:(t)l ( M whenever(t, $~(t),. . . , &(t))isintheregionR: It-TI 5 h, Ix, -2,I 5 k, (i = 1,. . . , n),whereMh 5 k, for i = 1, . . . , n:Show that (t, $l(t), . . . , $,(t)) is in R for It - il 5 h.9. Show that each complete solution of the differential equationdx- = t2arc tan x + e'd tis defined for -co < t < co. (See Problem 8(a).)Suggested ReferencesAgnew, Ralph P., Differential Equations, 2nd ed. New York: McGraw-Hill, 1960.Andronov, A. A., S. E. Khaikin, and A. A. Vitt, Theory of Oscillators. Reading, Mass.:Addison-Wesley, 1966.i