1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 323 Notice that this theo rem guarantees the existence of a unique solution on the ...

324 Ordinary Differential Equations If the initial conditions were specified at c = 2, then the interval of exis- tence and uniq ...

Systems of First-Order Differential Equations (19a ) U~-1 = Un (19b) EXAMPLE 3 Converting an n-th Order Initial Value Problem to ...

326 Ordinary Differential Equations interval about c provided in some generalized rectangle R in xu1 u2 ... Un- space, the funct ...

(21a) Systems of First-Order Differential Equations u' 1 U~-l = Un 327 Applying the fundamental existence and uniqueness theorem ...

328 Ordinary Differential Equations SOLUTION Letting u 1 = y, u 2 = y<^1 l , and u 3 = y<^2 l, we obtain the desired equiv ...

Systems of First-Order Differential Equations 329 SOLUTION Let u1 = x, u2 = x', u3 = y, and U4 = y'. So u 1 represents the x-coo ...

330 Ordinary Differential Equations Consider the system initial value problem (25a) 2y1 Y2 1 1 -3+-- - x x^2 x x^2 Y2 I 2 ...

Systems of First-Order Differential Equations Consider the system initial value problem (29a) (29b) I Y1 1 Y1 - 2Y1Y2 + -- x + ...

332 Ordinary Differential Equations Consider the system initial value problem (32a) (32b) 2./x + 4Y1Y2 - (sinx)y2 + 3ex -yr+ 4 ...

Systems of First-Order Differential Equations 333 For each of the following initial value problems determine the la rgest inter ...

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Chapter 8 Linear Systems of First-Order Differential Equations In this chapter we discuss linear systems of first-order differen ...

336 Ordinary Differential Equations DEFINITION Size of a Matrix If the matrix A h as m rows and n columns, we will write A is an ...

Linear Systems of First-Order Differential Equations The essential algebraic properties for matrices are stated below. DEFINITIO ...

338 Ordinary Differential Equations From the definition of addition, it is evident t hat matrix addition is com- mutative and as ...

Linear Systems of First-Order Differential Equations 339 For any matrix A , a A = A a. Hence, the following two distributive law ...

340 Ordinary Differential Equations following simple example illustrates. Let Then and A=(~ ~) and B = G ~). AB= (0 1 1) (1 0) ...

Linear Systems of First-Order Differential Equations 341 DEFINITION Determinant Let A be a square matrix of size n. The determin ...

342 and 2-.A 4 1 3 -1-.A 2 Ordinary Differential Equations -1 0 3-.A ( 2-.A =det ~ 3 -1-.A 2 -1 ) 3~.A = (2 - .A)(-1-.A)(3-.A)+ ...