Lecture Notes in Differential Equations - Bruce E. Shapiro

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328 LESSON 31. LINEAR SYSTEMS The corresponding eigenvectors are ( ) ( ) −1 1 v 1 = , v 1 2 = 1 The diagonalizing matrix is then and its inverse is given by so that S = (v 1 v 2 ) = S −1 = 1 2 D = S −1 PS = 1 2 ( −1 1 1 1 ( −1 1 1 1 ) ) ( ) −t 2 0 0 t 2 is diagonal. Hence M = e P = Se D S −1 , which we calculate as follows. ( ) ( ) ( ) −1 1 e M = −t2 /2 0 1 −1 1 1 1 0 e t2 /2 2 1 1 ( ) ( ) −1 1 −e −t2 /2 e −t2 /2 (31.209) (31.210) (31.211) (31.212) = 1 2 = 1 2 = 1 1 e t2 /2 e t2 /2 ( e −t2 /2 + e t2 /2 −e −t2 /2 + e t2 /2 −e −t2 /2 + e t2 /2 e −t2 /2 + e t2 /2 ( ) cosh(t 2 /2) sinh(t 2 /2) sinh(t 2 /2) cosh(t 2 /2) ) (31.213) and (using the identity cosh 2 x − sinh 2 x = 1), ( ) M −1 cosh(t = 2 /2) − sinh(t 2 /2) − sinh(t 2 /2) cosh(t 2 /2) Furthermore, ( ) ( ) cosh(t M(t)g(t) = 2 /2) sinh(t 2 /2) t sinh(t 2 /2) cosh(t 2 /2) 3t ( ) t cosh(t = 2 /2) + 3t sinh(t 2 /2) t sinh(t 2 /2) + 3t cosh(t 2 /2) (31.214) (31.215) (31.216) Using the integral formulas ∫ t cosh(t 2 /2)dt = sinh(t 2 /2) (31.217) ∫ t sinh(t 2 /2)dt = cosh(t 2 /2) (31.218)
329 we find that ∫ M(t)g(t)dt = ( ) 3 cosh(t 2 /2) + sinh(t 2 /2) cosh(t 2 /2) + 3 sinh(t 2 /2) (31.219) hence ∫ M −1 (t) M(t)g(t)dt (31.220) ( ) ( ) cosh(t = 2 /2) − sinh(t 2 /2) 3 cosh(t 2 /2) + sinh(t 2 /2) − sinh(t 2 /2) cosh(t 2 /2) cosh(t 2 /2) + 3 sinh(t 2 /2) (31.221) ( 3 = (31.222) 1) Since y = M −1 (t) [∫ M(t)g(t)dt + C ] , the general solution is ( ( ) ( ) 3 cosh(t y = + 1) 2 /2) − sinh(t 2 /2) C1 − sinh(t 2 /2) cosh(t 2 /2) C 2 ( ) 3 + C1 cosh(t = 2 /2) − C 2 sinh(t 2 /2) 1 − C 1 sinh(t 2 /2) + C 2 cosh(t 2 /2) (31.223) (31.224) The initial conditions give ( 4 2 ) = ( ) 3 + C1 1 + C 2 (31.225) Hence C 1 = C 2 = 1 and y = ( 3 + cosh(t 2 /2) − sinh(t 2 /2) 1 − sinh(t 2 /2) + cosh(t 2 /2) ) (31.226) Since cosh x − sinh x = ex + e −x 2 − ex − e −x 2 we have the solution of the initial value problem as y = ( 3 + e −t2 /2 1 + e −t2 /2 ) = e −x (31.227) . (31.228)
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Lecture Notes in Differential Equat
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Contents Front Cover . . . . . . .
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CONTENTS v Dedicated to the hundred
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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
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
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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
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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
Page 99 and 100:
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
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
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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
Page 133 and 134:
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 ≤ ‖
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 −
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159 and its roots are given by The
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161 The motivation for equation 18.
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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
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Lesson 22 Non-homogeneous Equations
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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
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197 The characteristic equation is
Page 207 and 208:
Lesson 23 Method of Annihilators In
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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.
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209 Example 24.3. Solve the initial
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Lesson 25 Harmonic Oscillations If
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213 It is standard to define a new
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215 As with the unforced case, we c
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Lesson 26 General Existence Theory*
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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 +
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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
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237 a closed form expression for a
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Page 249 and 250:
241 The characteristic equation is
Page 251 and 252:
243 The Wronskian In this section w
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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
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267 Table 28.1: Table of Special Fu
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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 and 324: 315 Theorem 31.8. (Abel’s Formula
Page 325 and 326: 317 By a similar argument, the seco
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: 327 Non-constant Coefficients We ca
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
Page 375 and 376: 367 result with a smaller step size
Page 377 and 378: 369 Expanding the final term in a T
Page 379 and 380: 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
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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
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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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