Lecture Notes in Differential Equations - Bruce E. Shapiro

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132 LESSON 15. LINEAR OPERATORS AND VECTOR SPACES Definition 15.4. If y, z are both elements of a vector space V, and A and B are any numbers, we call a Linear Combination of y and z w = Ay + Bz (15.41) Example 15.8. If v = (1, 0, 3) and w = (5, −3, 12) are vectors in Euclidean 3 space, then for any numbers A and B, u = Av + Bw = A(1, 0, 3) + B(5, −3, 12) = (A + 5B, −3B, 3A + 12B) (15.42) is a linear combination of v and w. The closure property of vector spaces is sometimes stated as following: Any linear combination of vectors is an element of the same vector space. For example, we can create linear combinations of functions and we know that they are also in the same vector space. Example 15.9. Let f(t) = 3 sin t, g(t) = t 2 − 4t be functions in the vector space V of real valued functions. Then if A and B are any real numbers, h(t) = Af(t) + Bg(t) (15.43) = A sin t + B(t 2 − 4t) (15.44) is a linear combination of the functions f and g. Since V is a vector space, h is also in V. Example 15.10. Let V be the vector space consisting of real functions on the real numbers. Then differentiation, defined by D(y) = dy(t) dt (15.45) is a linear operator. To see that both properties hold let y(t) and z(t) be functions (e.g., y, z ∈ V) and let c be a constant. Then D(y + z) = d(y(t) + z(t) dt = dy(t) dt D(cy) = d(cy(t) dt Hence D is a linear operator. + dz(t) dt = c dy(t) dt = D(y) + D(z) (15.46) = cD(y) (15.47)
133 Definition 15.5. Two vectors y, z are called linearly dependent if there exists nonzero constants A, B such that Ay + Bz = 0 (15.48) If no such A and B exist, then we say that y and z are Linearly Independent. In Euclidean 3D space, linearly dependent vectors are parallel, and linearly independent vectors can be extended into lines that will eventually intersect. Since we can define a vector space of functions, the following also definition makes sense. Definition 15.6. We say that two functions f and g are linearly dependent if there exist some nonzero constants such that Af(t) + Bg(t) = 0 (15.49) for all values of t. If no such constants exists then we say that f and g are linearly independent.
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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
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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
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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
Page 89 and 90: 81 Proof. In each of the five cases
Page 91 and 92: 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
Page 105 and 106: 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
Page 129 and 130: 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
Page 137 and 138: 129 Example 15.3. By a similar argu
Page 139: 131 Therefore ‖y + z‖ 2 ≤ ‖
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
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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
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189 where r 1 and r 2 are the roots
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191 This is a first order linear eq
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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
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Lesson 23 Method of Annihilators In
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201 Theorem 23.5. (D 2 − 2aD + (a
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203 The method of annihilators is r
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Lesson 24 Variation of Parameters T
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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
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223 Since K n /(1 − K) → 0 as n
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225 for any φ ∈ V. Let g, h be f
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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
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
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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
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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
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277 ∑ ∞ ∞∑ ∞∑ 0 = t 2 a
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279 Case 2: Two equal real roots. S
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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
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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
Page 331 and 332:
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 = (
Page 339 and 340:
Lesson 32 The Laplace Transform Bas
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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
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343 Derivatives of the Laplace Tran
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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
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355 Similarly, we can express a uni
Page 365 and 366:
357 Figure 32.7: Solution of exampl
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Lesson 33 Numerical Methods Euler
Page 369 and 370:
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
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
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
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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
Page 409 and 410:
401 ∫ ∫ ∫ ∫ e ax2 dx = −
Page 411 and 412:
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
Page 417 and 418:
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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