1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 203 and cos Ct where C is an arbitrary constant are linearly independent func- tions on ...

204 Ordinary Differential Equations W(y1, yz, ... , Yn, x) = A judicious choice for x in the Wronskian above is x = 0. Making th ...

N-th Order Linear Differential Equations 205 Factoring, we find p(r) = (r - 2)^3 = 0. The roots of this auxiliary equation are r ...

206 Ordinary Differential Equations Consequently, the set {y 1 ( x), y 2 ( x), y3 ( x)} is linearly independent on the in- terva ...

N-th Order Linear Differential Equations 207 Using the computer program POLYRTS, we find the roots of this polynomial are -3, -1 ...

208 Ordinary Differential Equations Each of these series converges for all x. Substituting ix for x in the expansion for ex, usi ...

N-th Order Linear Differential Equations 209 Furthermore, if the complex number a + i{3 where {3 ~ 0 and its complex conjugate a ...

210 Ordinary Differential Equations So the real, general solution to the given differential equation is y = c 1 e-x + c2e^3 x co ...

N-th Order Linear Differential Equations 211 c. Find the values of a which produce two equal real roots of the auxili ary equati ...

212 Ordinary Differential Equations functions on the interval ( -oo, oo), by the representation theorem for n-th order nonhomoge ...

N-th Order Linear Differential Equations 213 SOLUTION The associated homogeneous equation y^11 - 3y' + 2y = 0 has a uxiliary equ ...

214 Ordinary Differential Equations EXAMPLE 3 Wrong Initial Guess of the Form of the Particular Solution Solve the nonhomogeneou ...

N-th Order Linear Differential Equations 215 Substituting yp(x ) and its derivatives into equation (5), we find B must satisfy B ...

216 Ordinary Differential Equations where a , bm, bm-l, ... , b 1 , bo are constants and bm =/= 0, then there is a particular so ...

N-th Order Linear Differential Equations 217 EXAMPLE 4 General Solution of a Nonhomogeneous Differential Equation by Superpositi ...

218 Ordinary Differential Equations Since b 2 ( x) = -2e-^3 x is of the form (7) and r = -3 is a root of the auxiliary equation ...

N-th Order Linear Differential Equations 219 4.5 Initial Value Problems The initial value problem for a homogeneous linear diffe ...

220 Ordinary Differential Equations where an, an_ 1 , ... , a 1 , and ao are constants and where an -=/=- 0 subject to the initi ...

N-th Order Linear Differential Equations 221 SOLUTION Since the differential equation of this problem is equation (10) of sectio ...

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