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THE SELF-ADJOINT SECOND-ORDER DIFFERENTIAL EQUATION in Java Insert pdf417 2d barcode in Java THE SELF-ADJOINT SECOND-ORDER DIFFERENTIAL EQUATION

5. THE SELF-ADJOINT SECOND-ORDER DIFFERENTIAL EQUATION using barcode printer for javabean control to generate, create pdf 417 image in javabean applications. ISO Specification a given constant, is cal jar pdf417 2d barcode led a linear nonhomogeneous boundary condition, whereas any boundary condition of the form M x = is called a linear homogeneous boundary condition. If a boundary condition is not linear it is called a nonlinear boundary condition. Classify each of the following boundary conditions:.

(i) 2x(a) - 3x"(a) =. (ii) (iii) (iv) (v). x(a) x(a) x(a) x(a). + 6 = x(b) + 2x2(b) =. = x(b) =. (p(t)x")" x"(a). 5.54 Use Theorem 5.101 t o find the Green"s function for the left focal BVP 0,.

x(b). 5.55 Show that if the bo undary value problem (5.30)-(5.

32) has only the trivial solution, then the Green"s function for the BVP (5.30)-(5.32) satisfies G(s+, s) = G(s-, s) and satisfies the jump condition G"(s+, s) - G"(s-, s).

pts) ,. for s E (a, b). Here G(s +, s) := limt-+s+ G(t, s)..

5.56 Use Theorem 5.101 t jar pdf417 o find the Green"s function for the BVP x".

= 0,. x(o) - x"(O). = 0,. x(1). + x"(1) = 0.. 5.57 For each of the fol javabean PDF-417 2d barcode lowing, find an appropriate Green"s function and solve the given BVP:. (i) x" = t 2, x(O) = 0, x(1) = (ii) (e 2t x")" = e3t , x(O) = 0, x(log(2)) (iii) (c 5t x")" +6e- 5t x = e 3t , x(O) = 0,. x(log(2)). 5.58 Prove Corollary 5.1 02.

5.59 Use Corollary 5.102 to solve the BVPs x(O) = 1, x" (log(2)) = 2 (ii) x" + 4x = 5e t , x(O) = 1, x"(~) = (iii) t 2x" - 6tx" + 12x = 2t, x(1) = 0, x" (2) = 88.

(i) x" - 3x" + 2x = e 3t ,. 5.60 Let G( , ) be the G reen"s function for the right focal BVP x". Show that = 0,. x(a). = x"(b) = 0.. for ". -(b-a):::; G(t,s):::; 0,. a:::; t,s:::; b,. IG(t, s)1 ds :::;. a:::; t :::; b,. 5.10. EXERCISES lbIG"(t,S)lds:-=;b-a,. a:-=;t:-=;b. 5.61 In each of the foll jdk PDF-417 2d barcode owing use Theorem 5.113 to find T as large as possible so that the given differential equation is disconjugate on [0, T].

. (i) x". (ii) (iii) x". + t 2x = 0 x" + (t 2 + ~ + 3t 2 ). + (~ + 3t 2 ) X In part (iii) use your c alculator to solve the inequality in T that you got. 5.62 For each of the following, find an appropriate Green"s function and solve the given periodic BVP (note that by Theorem 5.

116 the solution you will find is periodic with period b - a):. (i) x" (ii) x" (iii) x". +x = +x = +x = x(O) x(O). = x(7r), = x(~),. x(O). cos(4t),. x"(O) = x"(7r) x"(O) = x pdf417 for Java "(~) xG), x"(O) = x"G). 6 . Linear Differential Equations of Order n 6.1 Basic Results In this chapter we are concerned with the nth-order linear differential equation yen). + Pn_l(t)y(n-l) + ... + Po(t)y = h(t),. (6.1). where we assume Pi : I -+ JR is continuous for 0 is continuous, where I is a subinterval of JR. n -1, and h : I Definition 6.1 Let C(1) := {x : I JR: x is continuous on I}. C n (1) := {x : 1-+ JR : PDF417 for Java x has n continuous derivatives on I},. and define Ln : C n (1). C (1) by Lnx(t) = x(n) (t). + Pn-l (t)x(n-l) (t) + . PDF-417 2d barcode for Java ..

+ Po(t)x(t),. for tEl. Then it can be shown (Exercise 6.1) that Ln is a linear operator.

Note that the differential equation (6.1) can be written in the form LnY = h. If h is not the trivial function on I, then LnY = h is said to be nonhomogeneous, whereas the differential equation Lnx = 0 is said to be homogeneous.

. Theorem 6.2 (Existence-U PDF-417 2d barcode for Java niqueness Theorem) Assume Pi : I -+ JR are continuous for 0 ~ i ~ n - 1, and h : I -+ JR is continuous. Assume that to E I and A E JR for 0 ::; i ::; n - 1.

Then the IVP. h(t),. y(i)(tO). 0::; i ::; n - 1,. (6.2). has a unique solution and this solution exists on the whole interval I. Proof The existence and jdk barcode pdf417 uniqueness of solutions of the IVP (6.2) follows from Corollary 8.19.

The fact that solutions exist on the whole interval I follows from Theorem 8.65. 0.

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