Lecture Notes in Differential Equations - Bruce E. Shapiro

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316 LESSON 31. LINEAR SYSTEMS Comparing the last two results, ∣ ∣ a 1 · y 1 a 1 · y 2 · · · a 1 · y n∣∣∣∣∣∣∣∣ y 11 y 21 · · · y n1 ∣∣∣∣∣∣∣∣ W ′ y 12 y 22 y n2 a 2 · y 1 a 2 · y 2 a 2 · y n (t) = . + . .. . . .. ∣ y 13 y 2n y nn ∣ y 13 y 2n y nn ∣ y 11 y 21 · · · y n1 ∣∣∣∣∣∣∣∣ y 12 y 22 + · · · + . (31.108) . .. ∣a n · y 1 a n · y 2 a n · y n The first determinant is ∣ a 1 · y 1 · · · a 1 · y n∣∣∣∣∣∣∣∣ y 12 y n2 (31.109) . . ∣ y 13 · · · y nn ∣ a 11 y 11 + · · · + a 1n y 1n a 11 y 21 + · · · + a 1n y 2n · · · a 11 y n1 + · · · + a 1n y nn∣∣∣∣∣∣∣∣ y 12 y 22 y n2 = . . .. ∣ y 13 y 2n y nn We can subtract a 21 times the second row, a 31 times the third row, ..., a n1 times the nth row from the first row without changing the value of the determinant, ∣ ∣ a 1 · y 1 a 1 · y 2 · · · a 1 · y n ∣∣∣∣∣∣∣∣ a 11 y 11 a 11 y 21 · · · a 11 y n1 ∣∣∣∣∣∣∣∣ y 12 y 22 y n2 y 12 y 22 y n2 . = . .. . . .. ∣ y 13 y 2n y nn ∣ y 13 y 2n y nn (31.110) We can factor out a 11 from every element in the first row, ∣ ∣ ∣ a 1 · y 1 a 1 · y 2 · · · a 1 · y n ∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣ y 11 y 21 · · · y n1 ∣∣∣∣∣∣∣∣ y 12 y 22 y n2 y 12 y 22 y n2 . = a . .. 11 . = a . .. 11 W (t) ∣ y 13 y 2n y nn y 13 y 2n y nn (31.111)
317 By a similar argument, the second determinant is a 22 W (t), the third one is a 33 W (t), ..., and the nth one is a nn W (t). Therefore W ′ (t) = (a 11 + a 22 + · · · + a nn )W (t) = (traceA)W (t) (31.112) Dividing by W (t) and integrating produces the desired result. Theorem 31.9. Let y 1 , ..., y n : I → R n be solutions of the linear system y ′ = Ay, where A is an n × n constant matrix, on some interval I ⊂ R, and let W (t) denote their Wronskian. Then the following are equivalent: 1. W (t) ≠ 0 for all t ∈ I. 2. For some t 0 ∈ I, W (t 0 ) ≠ 0 3. The set of functions y 1 (t), ..., y n (t) are linearly independent. 4. The set of functions y 1 (t), ..., y n (t) form a fundamental set of solutions to the system of differential equations on I. Proof. (1) ⇒ (2). Suppose W (t) ≠ 0 for all t ∈ I (this is (1)). Then pick any t 0 ∈ I. Then ∃t 0 ∈ I such that W (t 0 ) ≠ 0 (this is (2)). (2) ⇒ (1). This follows immediately from Abel’s formula. (1) ⇒ (3). Since the Wronskian is nonzero, the fundamental matrix is invertible. But a matrix is invertible if and only if its column vectors are linearly independent. (3) ⇒ (1). Since the column vectors of the fundamental matrix are linearly independent, this means the matrix is invertible, which in turn implies that its determinant is nonzero. (3) ⇒ (4). This was proven in a previous theorem. (4) ⇒ (3). Since the functions form a fundamental set, they must be linearly independent. Since (4) ⇐⇒ (3) ⇐⇒ (1) ⇐⇒ (2) all four statements are equivalent. Theorem 31.10. Let A be an n × n matrix with n linearly independent eigenvectors v 1 , ..., v n with corresponding eigenvalues λ 1 , ..., λ n . Then y 1 = v 1 e λ1t , . . . y n = v n e λnt (31.113) form a fundamental set of solutions for y ′ = Ay.
Page 1 and 2:
Lecture Notes in Differential Equat
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Contents Front Cover . . . . . . .
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CONTENTS v Dedicated to the hundred
Page 7 and 8:
Preface These lecture notes on diff
Page 9 and 10:
Lesson 1 Basic Concepts A different
Page 11 and 12:
3 Definition 1.2 (Solution, ODE). A
Page 13 and 14:
5 1.22 is restricted to being a pos
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7 Figure 1.1 illustrates what this
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9 We will study linear equations in
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Lesson 2 A Geometric View One way t
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13 We can extend this geometric int
Page 23 and 24:
15 that since the slope of the solu
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Lesson 3 Separable Equations An ODE
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19 Since it is not possible to solv
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21 where M(t) = −a(t) and N(y) =
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23 Example 3.10. Find a general sol
Page 33 and 34:
Lesson 4 Linear Equations Recall th
Page 35 and 36:
27 So far any function µ will work
Page 37 and 38:
29 Example 4.1. Solve the different
Page 39 and 40:
31 Since p(t) = 1 (the coefficient
Page 41 and 42:
33 Multiplying equation 4.68 by µ
Page 43 and 44:
35 Substituting y = t = 0 in this i
Page 45 and 46:
37 ∫ t t 0 Evaluating the integra
Page 47 and 48:
Lesson 5 Bernoulli Equations The Be
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41 This is a Bernoulli equation wit
Page 51 and 52:
Lesson 6 Exponential Relaxation One
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45 Exponential Runaway First we con
Page 55 and 56:
47 Figure 6.2: Illustration of the
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49 This is identical to with Theref
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51 this becomes a first-order ODE i
Page 61 and 62:
Lesson 7 Autonomous Differential Eq
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55 Figure 7.1: A plot of the right-
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57 Figure 7.2: Solutions of the log
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59 Figure 7.4: Solutions of the thr
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Lesson 8 Homogeneous Equations Defi
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63 where z = y/t, the differential
Page 73 and 74:
Lesson 9 Exact Equations We can re-
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67 Now compare equation (9.2) with
Page 77 and 78:
69 Hence dg dy = 0 =⇒ g = C′ (9
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71 From the first of equations (9.5
Page 81 and 82:
73 Differentiating equations (9.81)
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75 This has the form Mdt + Ndy = 0
Page 85 and 86:
Lesson 10 Integrating Factors Defin
Page 87 and 88:
79 Differentiating with respect to
Page 89 and 90:
81 Proof. In each of the five cases
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83 as required by equation (10.31).
Page 93 and 94:
85 Since M y ≠ N t , equation (10
Page 95 and 96:
87 the revised equation (10.100) is
Page 97 and 98:
89 Substituting (10.129) into (10.1
Page 99 and 100:
Lesson 11 Method of Successive Appr
Page 101 and 102:
93 because the integral is zero (th
Page 103 and 104:
95 Example 11.1. Construct the Pica
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97 We can then plug this expression
Page 107 and 108:
Lesson 12 Existence of Solutions* I
Page 109 and 110:
101 • Interchangeability of Limit
Page 111 and 112:
103 But on the square −1 ≤ t
Page 113 and 114:
105 Thus lim φ n = φ 0 + lim n→
Page 115 and 116:
107 because the right hand side doe
Page 117 and 118:
Lesson 13 Uniqueness of Solutions*
Page 119 and 120:
111 The proof of theorem (13.1) is
Page 121 and 122:
113 But δ(t) is an absolute value,
Page 123 and 124:
115 Substituting (13.66) into (13.6
Page 125 and 126:
Lesson 14 Review of Linear Algebra
Page 127 and 128:
119 Definition 14.10. An m × n (or
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121 Definition 14.19. Matrix Multip
Page 131 and 132:
123 In practical terms, computation
Page 133 and 134:
125 Simplifying 4x − 2 + 3z = 0 (
Page 135 and 136:
Lesson 15 Linear Operators and Vect
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129 Example 15.3. By a similar argu
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131 Therefore ‖y + z‖ 2 ≤ ‖
Page 141 and 142:
133 Definition 15.5. Two vectors y,
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Lesson 16 Linear Equations With Con
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137 Hence both r = 1 and r = 3. Thi
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139 The second order linear initial
Page 149 and 150:
141 The general solution to is give
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Lesson 17 Some Special Substitution
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145 Therefore since z = y ′ , Int
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147 Example 17.5. Solve yy ′′ +
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149 where I is the identity matrix.
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151 can be rewritten by solving a =
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Lesson 18 Complex Roots We know for
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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
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161 The motivation for equation 18.
Page 171 and 172:
Lesson 19 Method of Undetermined Co
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165 3. If f(t) = e rt and r is a ro
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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 and 194:
185 Example 21.5. Find a second sol
Page 195 and 196:
Lesson 22 Non-homogeneous Equations
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
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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
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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
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221 Theorem 26.5. Under the same co
Page 231 and 232:
223 Since K n /(1 − K) → 0 as n
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225 for any φ ∈ V. Let g, h be f
Page 235 and 236:
Lesson 27 Higher Order Linear Equat
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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
Page 245 and 246:
237 a closed form expression for a
Page 247 and 248:
Page 249 and 250:
241 The characteristic equation is
Page 251 and 252:
243 The Wronskian In this section w
Page 253 and 254:
245 Certainly every φ(t) given by
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247 the differential equation. Over
Page 257 and 258:
249 By the lemma, to obtain the der
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Page 261 and 262:
253 So that f(t) = a n (t)y[ (n) +
Page 263 and 264:
Lesson 28 Series Solutions In many
Page 265 and 266:
257 Changing the index of the secon
Page 267 and 268:
259 Since the first two terms (corr
Page 269 and 270:
261 Hence ∞∑ ∞∑ ∞∑ 0 =
Page 271 and 272:
263 has an analytic solution at t =
Page 273 and 274: 265 By the triangle inequality, |(k
Page 275 and 276: 267 Table 28.1: Table of Special Fu
Page 277 and 278: 269 Thus y = a 0 ( 1 + 1 6 t3 + 1 +
Page 279 and 280: 271 into (28.114) and collect terms
Page 281 and 282: 273 Summary of Power series method.
Page 283 and 284: Lesson 29 Regular Singularities The
Page 285 and 286: 277 ∑ ∞ ∞∑ ∞∑ 0 = t 2 a
Page 287 and 288: 279 Case 2: Two equal real roots. S
Page 289 and 290: 281 Example 29.6. Solve t 2 y ′
Page 291 and 292: Lesson 30 The Method of Frobenius I
Page 293 and 294: 285 This is a homogeneous linear eq
Page 295 and 296: 287 Example 30.4. Find a Frobenius
Page 297 and 298: 289 Thus a Frobenius solution is y
Page 299 and 300: 291 Example 30.6. Find the form of
Page 301 and 302: 293 term by term to (30.97). Starti
Page 303 and 304: 295 Let j = n − k. Then |n − 1
Page 305 and 306: 297 is a solution of (t − t 0 ) 2
Page 307 and 308: 299 Evaluation of the integral depe
Page 309 and 310: 301 Example 30.8. In example 30.4 w
Page 311 and 312: Lesson 31 Linear Systems The genera
Page 313 and 314: 305 is where λ 2 − T λ + ∆ =
Page 315 and 316: 307 (b) If λ 1 ≠ λ 2 ∈ R, i.e
Page 317 and 318: 309 We will verify (31.54) by induc
Page 319 and 320: 311 The Jordan Form Let A be a squa
Page 321 and 322: 313 y = 1 [( ) ( ) ] ( ) 4 1 e 2t 1
Page 323: 315 Theorem 31.8. (Abel’s Formula
Page 327 and 328: 319 we can replace (31.118) with a
Page 329 and 330: 321 Corollary 31.12. The generalize
Page 331 and 332: 323 where (A − λI)w 2 = w 1 , i.
Page 333 and 334: 325 so that λ = 3, −5. The eigen
Page 335 and 336: 327 Non-constant Coefficients We ca
Page 337 and 338: 329 we find that ∫ M(t)g(t)dt = (
Page 339 and 340: Lesson 32 The Laplace Transform Bas
Page 341 and 342: 333 Figure 32.1: A piecewise contin
Page 343 and 344: 335 Example 32.4. From integral A.1
Page 345 and 346: 337 apply this result iteratively.
Page 347 and 348: L [ t x−1] [ ] 1 d = L x dt tx =
Page 349 and 350: 341 Equating numerators and expandi
Page 351 and 352: 343 Derivatives of the Laplace Tran
Page 353 and 354: 345 can be written as as illustrate
Page 355 and 356: 347 Translations in the Laplace Var
Page 357 and 358: 349 Summary of Translation Formulas
Page 359 and 360: 351 The inverse transform is [ ] f(
Page 361 and 362: 353 Example 32.18. Find the Laplace
Page 363 and 364: 355 Similarly, we can express a uni
Page 365 and 366: 357 Figure 32.7: Solution of exampl
Page 367 and 368: Lesson 33 Numerical Methods Euler
Page 369 and 370: 361 Figure 33.1: Illustration of Eu
Page 371 and 372: 363 y 4 = y 3 + hf(t 3 , y 3 ) (33.
Page 373 and 374: 365 Figure 33.3: Illustration of th
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367 result with a smaller step size
Page 377 and 378:
369 Expanding the final term in a T
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371 k 2 = y 0 + h 2 f(t 0, k 1 ) (3
Page 381 and 382:
Lesson 34 Critical Points of Autono
Page 383 and 384:
375 Since both f and g are differen
Page 385 and 386:
377 Using the cos π/4 = √ 2/2 an
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379 values, of the matrix. We find
Page 389 and 390:
381 Distinct Real Nonzero Eigenvalu
Page 391 and 392:
383 eigendirection {λ 1 , v 1 }dom
Page 393 and 394:
385 Figure 34.5: Phase portraits ty
Page 395 and 396:
387 Complex Conjugate Pair with non
Page 397 and 398:
389 The angular change is described
Page 399 and 400:
391 Figure 34.8: Topological instab
Page 401 and 402:
393 Figure 34.10: phase portraits f
Page 403 and 404:
Appendix A Table of Integrals Basic
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397 ∫ x √ x − adx = 2 3 a(x
Page 407 and 408:
399 ∫ x √ ax2 + bx + c dx = 1 a
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401 ∫ ∫ ∫ ∫ e ax2 dx = −
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403 ∫ tan 3 axdx = 1 a ln cos ax
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405 Products of Trigonometric Funct
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Appendix B Table of Laplace Transfo
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409 e at cosh kt t sin kt t cos kt
Page 419 and 420:
Appendix C Summary of Methods First
Page 421 and 422:
413 The resulting equation is linea
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415 for y once z is known. Method o
Page 425 and 426:
Bibliography [1] Bear, H.S. Differe
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BIBLIOGRAPHY 419
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BIBLIOGRAPHY 421
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