Lecture Notes in Differential Equations - Bruce E. Shapiro
Lecture Notes in Differential Equations - Bruce E. Shapiro Lecture Notes in Differential Equations - Bruce E. Shapiro
186 LESSON 21. REDUCTION OF ORDER
Lesson 22 Non-homogeneous Equations with Constant Coefficients Theorem 22.1. Existence and Uniqueness. The second order linear initial value problem a(t)y ′′ + b(t)y ′ + c(t)y = f(t) y(t 0 ) = y 0 y(t 0 ) = y 1 ⎫ ⎪⎬ ⎪ ⎭ (22.1) has a unique solution, except possibly where a(t) = 0. In particular, the second order linear initial value with constant coefficients, has a unique solution. ay ′′ + by ′ + cy = f(t) y(t 0 ) = y 0 y(t 0 ) = y 1 ⎫ ⎪⎬ ⎪ ⎭ (22.2) We will omit the proof of this for now, since it will follow as an immediate consequence of the more general result for systems we will prove in chapter (26). Theorem 22.2. Every solution of the differential equation ay ′′ + by ′ + cy = f(t) (22.3) 187
- Page 143 and 144: Lesson 16 Linear Equations With Con
- Page 145 and 146: 137 Hence both r = 1 and r = 3. Thi
- Page 147 and 148: 139 The second order linear initial
- Page 149 and 150: 141 The general solution to is give
- Page 151 and 152: Lesson 17 Some Special Substitution
- Page 153 and 154: 145 Therefore since z = y ′ , Int
- Page 155 and 156: 147 Example 17.5. Solve yy ′′ +
- Page 157 and 158: 149 where I is the identity matrix.
- Page 159 and 160: 151 can be rewritten by solving a =
- Page 161 and 162: Lesson 18 Complex Roots We know for
- Page 163 and 164: 155 Theorem 18.2. Euler’s Formula
- Page 165 and 166: 157 For k = 0, 1, 2, . . . , n −
- Page 167 and 168: 159 and its roots are given by The
- Page 169 and 170: 161 The motivation for equation 18.
- Page 171 and 172: Lesson 19 Method of Undetermined Co
- Page 173 and 174: 165 3. If f(t) = e rt and r is a ro
- Page 175 and 176: 167 Example 19.4. Solve ⎫ y ′
- Page 177 and 178: 169 Adding the two equations gives
- Page 179 and 180: Lesson 20 The Wronskian We have see
- Page 181 and 182: 173 Definition 20.1. The Wronskian
- Page 183 and 184: 175 Example 20.3. Show that y = sin
- Page 185 and 186: 177 and therefore the system of equ
- Page 187 and 188: Lesson 21 Reduction of Order The me
- Page 189 and 190: 181 The method of reduction of orde
- Page 191 and 192: 183 Plugging these into Bessel’s
- Page 193: 185 Example 21.5. Find a second sol
- Page 197 and 198: 189 where r 1 and r 2 are the roots
- Page 199 and 200: 191 This is a first order linear eq
- Page 201 and 202: 193 Theorem 22.5. Properties of the
- Page 203 and 204: 195 where (∫ ν(t) = exp ) −r 2
- Page 205 and 206: 197 The characteristic equation is
- Page 207 and 208: Lesson 23 Method of Annihilators In
- Page 209 and 210: 201 Theorem 23.5. (D 2 − 2aD + (a
- Page 211 and 212: 203 The method of annihilators is r
- Page 213 and 214: Lesson 24 Variation of Parameters T
- Page 215 and 216: 207 Substituting into equation (24.
- Page 217 and 218: 209 Example 24.3. Solve the initial
- Page 219 and 220: Lesson 25 Harmonic Oscillations If
- Page 221 and 222: 213 It is standard to define a new
- Page 223 and 224: 215 As with the unforced case, we c
- Page 225 and 226: Lesson 26 General Existence Theory*
- Page 227 and 228: 219 In the case just proven, there
- Page 229 and 230: 221 Theorem 26.5. Under the same co
- Page 231 and 232: 223 Since K n /(1 − K) → 0 as n
- Page 233 and 234: 225 for any φ ∈ V. Let g, h be f
- Page 235 and 236: Lesson 27 Higher Order Linear Equat
- Page 237 and 238: 229 L n+1 (e rt y) = e rt a n (D +
- Page 239 and 240: 231 Example 27.2. Find the general
- Page 241 and 242: 233 Differentiating, u ′ (t) = d
- Page 243 and 244: 235 Integrating, − 2K |t − t 0
Lesson 22<br />
Non-homogeneous<br />
<strong>Equations</strong> with Constant<br />
Coefficients<br />
Theorem 22.1. Existence and Uniqueness. The second order l<strong>in</strong>ear<br />
<strong>in</strong>itial value problem<br />
a(t)y ′′ + b(t)y ′ + c(t)y = f(t)<br />
y(t 0 ) = y 0<br />
y(t 0 ) = y 1<br />
⎫<br />
⎪⎬<br />
⎪ ⎭<br />
(22.1)<br />
has a unique solution, except possibly where a(t) = 0. In particular, the<br />
second order l<strong>in</strong>ear <strong>in</strong>itial value with constant coefficients,<br />
has a unique solution.<br />
ay ′′ + by ′ + cy = f(t)<br />
y(t 0 ) = y 0<br />
y(t 0 ) = y 1<br />
⎫<br />
⎪⎬<br />
⎪ ⎭<br />
(22.2)<br />
We will omit the proof of this for now, s<strong>in</strong>ce it will follow as an immediate<br />
consequence of the more general result for systems we will prove <strong>in</strong> chapter<br />
(26).<br />
Theorem 22.2. Every solution of the differential equation<br />
ay ′′ + by ′ + cy = f(t) (22.3)<br />
187
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