Lecture Notes in Differential Equations - Bruce E. Shapiro

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232 LESSON 27. HIGHER ORDER EQUATIONS To prove theorem 27.3 1 we will need a generalization of the fundamental identity. We first define a the following norm of a function: n−1 ∑ ‖f(t)‖ 2 ∣ = ∣f (i) (t) ∣ 2 (27.35) i=0 = |f(t)| + |f ′ ∣ (t)| + · · · + ∣f (n−1) (t) ∣ 2 (27.36) That ‖· · · ‖ defines a norm follows from the fact that for any two functions f(t) and g(t), and for and constant c, the following four properties are true for all t: Properties of a Norm of a Function: 1. ‖f(t)‖ ≥ 0 2. ‖f(t)‖ = 0 ⇔ f(t) = 0 3. ‖f(t) + g(t)‖ ≤ ‖f(t)‖ + ‖g(t)‖ 4. ‖cf(t)‖ ≤ |c| ‖f(t)‖ For comparison to the norm of a vector space, see definition 15.2 which is equivalent to this for the vector space of functions. Lemma 27.5. (Fundamental Identity for nth order equations) Let φ(t) be a solution of L n y = 0, y(t 0 ) = y 0 , ..., y (n−1) (t 0 ) = y n (27.37) then for all t. ‖φ(t 0 )‖ e −K‖t−t0‖ ≤ ‖φ(t)‖ ≤ ‖φ(t 0 )‖ e K‖t−t0‖ (27.38) Lemma 27.6. 2 |a| |b| ≤ |a| 2 + |b| 2 Proof. (|a| + |b|) 2 = |a| 2 + |b| 2 − 2 |a| |b| ≥ 0. Proof. (of Lemma 27.5.) We can assume that a n ≠ 0; otherwise this would not be an nth-order equation. Further, we will assume that a n = 1; otherwise, redefine L n by division through by a n . Let n−1 ∑ u(t) = ‖φ(t)‖ 2 ∣ = ∣φ (i) (t) ∣ 2 n−1 ∑ = φ (i) (t)φ (i)∗ (t) (27.39) i=0 1 This material is somewhat more abstract and the reader may wish to skip ahead to the examples following the proofs. i=0
233 Differentiating, u ′ (t) = d n−1 ∑ ∣ ∣φ (i) (t) ∣ 2 = d n−1 ∑ φ (i) (t)φ (i)∗ (t) (27.40) dt dt i=0 Taking the absolute value and applying the triangle inequality twice Since |a| = |a ∗ | n−1 ∑ |u ′ ∣ (t)| ≤ ∣φ (i+1) (t)φ (i)∗ (t) + φ (i) (t)φ (i+1)∗ (t) ∣ (27.41) i=0 i=0 n−1 ∑ |u ′ ∣ (t)| ≤ 2 ∣φ (i+1) ∣ (t) ∣ ∣φ (i) (t) ∣ (27.42) i=0 Isolating the highest order term { n−2 } ∑ |u ′ ∣ (t)| ≤ 2 ∣φ (i+1) ∣ (t) ∣ ∣φ (i) ∣ (t) ∣ + 2 ∣φ (n) ∣ (t) ∣ ∣φ (n−1) (t) ∣ (27.43) i=0 Since L n φ(t) = 0 and a n = 1, n−1 ∣ ∣φ (n) (t) ∣ = ∣ − ∑ n−1 a i φ (i) (t) ∣ ≤ ∑ n−1 ∣ ∣a i φ (i) ∑ ∣ (t) ∣ = |a i | ∣φ (i) (t) ∣ (27.44) i=0 i=0 Combining the last two results, { n−2 } ∑ n−1 |u ′ ∣ (t)| ≤ 2 ∣φ (i+1) ∣ (t) ∣ ∣φ (i) ∑ ∣ (t) ∣ + 2 |a ∣ i | ∣φ (i) ∣ (t) ∣ ∣φ (n−1) (t) ∣ ∣ i=0 i=0 (27.45) By lemma 2, ∣ ∣ ∣ ∣ 2 ∣φ (i+1) ∣∣ ∣∣φ (i) ∣ ∣ ≤ ∣φ (i+1) ∣∣ 2 ∣ ∣∣φ + (i) ∣ 2 (27.46) Hence |u ′ (t)| ≤ i=0 { n−2 ∑ ( ∣∣∣φ (i+1) (t)∣ ∣ 2 ∣ + ∣φ (i) (t) ∣ 2)} i=0 n−1 ∣ +2 ∣φ (n−1) ∑ ∣ (t) ∣ |a i | ∣φ (i) (t) ∣ (27.47) i=0 By a change of index in the first term (let j = i + 1) n−2 ∑ ∣ ∣ ∣φ (i+1) ∣∣ 2 n−1 ∑ ∣ = ∣φ (j) (t) ∣ 2 (27.48) i=0 j=1
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
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3 Definition 1.2 (Solution, ODE). A
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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
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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
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Lesson 4 Linear Equations Recall th
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27 So far any function µ will work
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29 Example 4.1. Solve the different
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31 Since p(t) = 1 (the coefficient
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33 Multiplying equation 4.68 by µ
Page 43 and 44:
35 Substituting y = t = 0 in this i
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37 ∫ t t 0 Evaluating the integra
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Lesson 5 Bernoulli Equations The Be
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41 This is a Bernoulli equation wit
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Lesson 6 Exponential Relaxation One
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45 Exponential Runaway First we con
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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
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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
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Lesson 9 Exact Equations We can re-
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67 Now compare equation (9.2) with
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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
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Lesson 10 Integrating Factors Defin
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79 Differentiating with respect to
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81 Proof. In each of the five cases
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83 as required by equation (10.31).
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85 Since M y ≠ N t , equation (10
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87 the revised equation (10.100) is
Page 97 and 98:
89 Substituting (10.129) into (10.1
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Lesson 11 Method of Successive Appr
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93 because the integral is zero (th
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95 Example 11.1. Construct the Pica
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97 We can then plug this expression
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Lesson 12 Existence of Solutions* I
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101 • Interchangeability of Limit
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103 But on the square −1 ≤ t
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105 Thus lim φ n = φ 0 + lim n→
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107 because the right hand side doe
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Lesson 13 Uniqueness of Solutions*
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111 The proof of theorem (13.1) is
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113 But δ(t) is an absolute value,
Page 123 and 124:
115 Substituting (13.66) into (13.6
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Lesson 14 Review of Linear Algebra
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119 Definition 14.10. An m × n (or
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121 Definition 14.19. Matrix Multip
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123 In practical terms, computation
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125 Simplifying 4x − 2 + 3z = 0 (
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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 ≤ ‖
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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
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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
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157 For k = 0, 1, 2, . . . , n −
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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
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173 Definition 20.1. The Wronskian
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175 Example 20.3. Show that y = sin
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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
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: 231 Example 27.2. Find the general
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: 239 Example 27.6. Find the general
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
Page 255 and 256: 247 the differential equation. Over
Page 257 and 258: 249 By the lemma, to obtain the der
Page 259 and 260: 251 Example 27.14. Find the general
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 ′
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Lesson 30 The Method of Frobenius I
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285 This is a homogeneous linear eq
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287 Example 30.4. Find a Frobenius
Page 297 and 298:
289 Thus a Frobenius solution is y
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291 Example 30.6. Find the form of
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293 term by term to (30.97). Starti
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295 Let j = n − k. Then |n − 1
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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
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Lesson 31 Linear Systems The genera
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305 is where λ 2 − T λ + ∆ =
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307 (b) If λ 1 ≠ λ 2 ∈ R, i.e
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309 We will verify (31.54) by induc
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311 The Jordan Form Let A be a squa
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313 y = 1 [( ) ( ) ] ( ) 4 1 e 2t 1
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315 Theorem 31.8. (Abel’s Formula
Page 325 and 326:
317 By a similar argument, the seco
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319 we can replace (31.118) with a
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321 Corollary 31.12. The generalize
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323 where (A − λI)w 2 = w 1 , i.
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325 so that λ = 3, −5. The eigen
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327 Non-constant Coefficients We ca
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329 we find that ∫ M(t)g(t)dt = (
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Lesson 32 The Laplace Transform Bas
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333 Figure 32.1: A piecewise contin
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335 Example 32.4. From integral A.1
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337 apply this result iteratively.
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L [ t x−1] [ ] 1 d = L x dt tx =
Page 349 and 350:
341 Equating numerators and expandi
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343 Derivatives of the Laplace Tran
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345 can be written as as illustrate
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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(
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353 Example 32.18. Find the Laplace
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355 Similarly, we can express a uni
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357 Figure 32.7: Solution of exampl
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Lesson 33 Numerical Methods Euler
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361 Figure 33.1: Illustration of Eu
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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
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375 Since both f and g are differen
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377 Using the cos π/4 = √ 2/2 an
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379 values, of the matrix. We find
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381 Distinct Real Nonzero Eigenvalu
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383 eigendirection {λ 1 , v 1 }dom
Page 393 and 394:
385 Figure 34.5: Phase portraits ty
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387 Complex Conjugate Pair with non
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389 The angular change is described
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391 Figure 34.8: Topological instab
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393 Figure 34.10: phase portraits f
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Appendix A Table of Integrals Basic
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397 ∫ x √ x − adx = 2 3 a(x
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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
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Appendix C Summary of Methods First
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413 The resulting equation is linea
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415 for y once z is known. Method o
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Bibliography [1] Bear, H.S. Differe
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BIBLIOGRAPHY 419
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BIBLIOGRAPHY 421
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Lecture Notes in Differential Equations - Bruce E. Shapiro

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