1550078481-Ordinary_Differential_Equations__Roberts_

Chapter 4 N-th Order Linear Differential Equations As we noted earlier, ordinary differential equations are divided into two dis ...

164 Ordinary Differential Equations nonhomogeneous linear differential equation and 4yC^2 ) - 3xy = 0 is a second- order homogen ...

N-th Order Linear Differential Equations 165 DEFINITION Initial Value Problem An initial value problem for an n-th order linear ...

166 Ordinary Differential Equations on any interval on which the functions an(x), an-i(x), ... , a 1 (x), ao(x) are all continuo ...

N-th Order Linear Differential Equations 167 The function ( 5) y( x) = 2ex - 3e^2 x + 2x^2 + 6x + 7 is defined and continuous on ...

168 and Ordinary Differential Equati ons 6 y"(x) = 4 + 4. x Substituting these expressions for y , y' , and y" into t he DE (7), ...

N-th Order Linear Differential Equations 169 Since Y2, y~, and Y2 are defined on ( - oo, oo ), Y2 = ex is a solution of the DE ( ...

170 Ordinary Differential Equations Since y 1 (x) is a solution of (12) on I , a3(x)y~' + a2(x)y~ + a 1 (x)y~ + ao(x)y 1 = 0 for ...

N-th Order Linear Differential Equations 171 DEFINITIONS Linearly Dependent and Linearly Independent Let {YI ( x), Y2 ( x), ... ...

172 Ordinary Differential Equations C1Y1(Xm) + C2Y2(Xm) + · · · + CmYm(Xm) = 0 is C1 = C2 = ... = Cm = 0. In order to prove that ...

N-th Order Linear Differential Equations 173 THEOREM 4.1 Let the functions y 1 (x),y 2 (x), ... ,ym(x) all be differen- tiable a ...

174 Ordinary Differential Equations = sin 2x( - sin^2 x + cos x^2 ) - 2 cos 2x(sin x cos x) = (2sinxcosx)(cos2x) - 2sinxcosxcos2 ...

N-th Order Linear Differential Equations 175 Therefore, by Theorem 4.2, the functions 1, x, and x^2 are linearly independent on ...

176 Ordinary Differential Equations THEOREM 4.3 Let an(x), an-i(x), ... , a 1 (x), ao(x) be continuous on the interval I and let ...

N-th Order Linear Differential Equations 177 zero function. Hence, y(x) = k1y1(x) + kzyz(x) + · · · + knYn(x) = 0 for all x EI. ...

178 Ordinary Differential Equations 1 0 0 W(ex,xex,x^2 ex,0)= 1 1 0 =2/=0. 1 2 2 So the functions ex, xex, and x^2 ex are linear ...

N-th Order Linear Differential Equations 179 EXISTENCE OF N LINEARLY INDEPENDENT SOLUTIONS TON-TH ORDER HOMOGENEOUS LINEAR DIFFE ...

180 Ordinary Differential Equations By Theorem 4.4 there are at least n linearly independent solutions to the DE (19) on the int ...

N-th Order Linear Differential Equations 181 From linear algebra, we know that this system of equations has a unique so- lution ...

182 Ordinary Differential Equations DEFINITIONS Nonhomogeneous Linear DE and Associated Homogeneous Linear DE The general n-th o ...