1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 3 By 1670 the idea of a limit had been conceived; integration had been de- fined; many integrals had been calculate ...

4 Ordinary Differential Equations Sometime b etween 1673 and 1676 Leibniz invented his calculus. In 1675 he introduced the moder ...

Introduction 5 tures, or (iii) to compute a n approximate solution. Early in the development of the subject of differential equ ...

6 Ordinary Differential Equati ons four roots to this equation and they can all b e found in the set of complex numbers. This la ...

Introduction 7 contains partial derivatives, then the differential equation is called a partial differential equation (PDE). The ...

8 Ordinary Differential Equations we introduce another notation, y(^2 ), for the second derivative. The tenth derivative in this ...

Introduction 9 Functions of y or any of its derivatives such as eY or cosy' cannot appear in the equation. Equations (1), (2), ...

10 Ordinary Differential Equations I EXAMPLE 1 Verification of a Solution Verify that y = xe^2 x is a so lution of the linear se ...

Introduction 11 1 So the function y = x + - is a solution of x^2 y" + xy^1 -y = 0 on ( -oo, 0), x on (0, oo ), and on any subint ...

12 Ordinary Differential Equations For example, the piecewise defined function { x^3 + 2, y( x)= x2-1, x<O 0:::; x is defined ...

Introduction I EXAMPLE 3 A Solution D e fined Piecewise Verify that the piecewise defined function y( x ) = - 2 ' { x 2 x, x < ...

14 Ordinary Differential Equations For x < 0, xy' - 2y = x(-2x) - 2(-x^2 ) = -2x^2 + 2x^2 = 0, so y(x) satisfies the given di ...

Introduction 15 In exercises 18-26 v erify that the given function or functions is a solution of the give n diffe r ential equat ...

16 Ordinary Differential Equations Verify that the piecewise defined function { 0, y(x) = x2 ' x<O 0 :::: x is differentiab ...

Introduction 17 1.3 Solutions and Problems In the previous section , we learned what a solution to a differential equ a- tion is ...

18 Ordinary Differential Equations Integrating the second-order differential equation y" = 2x + 1 twice, we find y' = x^2 + x + ...

Introduction 19 DEFINITION General Solution and Singular Solution If every solution on an interval I of the n-th order different ...

20 Ordinary Differential Equations b. Show that y(x) = -x^2 /4 is a singular solution of the DE (8). SOLUTION a. Differentiating ...

Introduction 21 Given a function y( x), we are usually able to determine where the function is differentiable and to calculate i ...

22 Ordin ary Differential Equations DEFINITION Initia l Value Problem The problem of finding a solution y = f(x ) of the n-th or ...