Lecture Notes in Differential Equations - Bruce E. Shapiro

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142 LESSON 16. LINEAR EQS. W/ CONST. COEFFICENTS Example 16.5. Solve the initial value problem ⎫ y ′′ − 6y ′ + 9y = 0 ⎪⎬ y(0) = 4 ⎪⎭ y ′ (0) = 17 (16.71) The characteristic equation is 0 = r 2 − 6r + 9 = (r − 3) 2 (16.72) Since there is a repeated root r = 3, the general solution of the homogeneous equation is y = Ae 3t + Bte 3t (16.73) By the first initial condition, we have 4 = y(0) = A.Hence Differentiating, From the second initial condition, y = 4e 3t + Bte 3t (16.74) y ′ = 12e 3t + Be et + 3Bte 3t (16.75) 17 = y ′ (0) = 12 + B + 3B = 12 + B =⇒ B = 5 (16.76) Hence the solution of the initial value problem is y = (4 + 5t)e 3t . (16.77)
Lesson 17 Some Special Substitutions Equations with no y dependence. If c = 0 in L then the differential equation simplifies to Ly = f(t) (17.1) ay ′′ + by ′ = f(t) (17.2) In this case it is possible to solve the equation by making the change of variables z = y ′ , which reduces the ODE to a first order linear equation in z. This works even when a or b have t dependence. This method is illustrated in the following example. Example 17.1. Find the general solution of the homogeneous linear equation y ′′ + 6y ′ = 0 (17.3) Making the substitution z = y ′ in (17.3) gives us z ′ + 6z = 0 (17.4) Separating variables and integrating gives ∫ ∫ dz z = − 6dt =⇒ z = Ce −6t (17.5) 143
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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
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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
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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 and 140: 131 Therefore ‖y + z‖ 2 ≤ ‖
Page 141 and 142: 133 Definition 15.5. Two vectors y,
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: 141 The general solution to is give
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 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
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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
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
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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
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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
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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
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249 By the lemma, to obtain the der
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253 So that f(t) = a n (t)y[ (n) +
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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 =
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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 +
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271 into (28.114) and collect terms
Page 281 and 282:
273 Summary of Power series method.
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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 ′
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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
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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 =
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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
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349 Summary of Translation Formulas
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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.
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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
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Lesson 34 Critical Points of Autono
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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
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383 eigendirection {λ 1 , v 1 }dom
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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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