Lecture Notes in Differential Equations - Bruce E. Shapiro
Lecture Notes in Differential Equations - Bruce E. Shapiro Lecture Notes in Differential Equations - Bruce E. Shapiro
134 LESSON 15. LINEAR OPERATORS AND VECTOR SPACES
Lesson 16 Linear Equations With Constant Coefficients Definition 16.1. The general second order linear equation with constant coefficients is ay ′′ + by ′ + cy = f(t) (16.1) where a, b, and c are constants, and a ≠ 0 (otherwise (16.1) reduces to a linear first order equation, which we have already covered), and f(t) depends only on t and not on y. Definition 16.2. The Linear Differential Operator corresponding to equation (16.1) L = aD 2 + bD + c (16.2) where D = d dt and D2 = d2 dt 2 (16.3) is the same operator we introduced in example (15.10). We can also write L as L = a d2 dt 2 + b d dt + c (16.4) In terms of the operator L, equation (16.1) becomes Ly = f(t) (16.5) 135
- 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 and 140: 131 Therefore ‖y + z‖ 2 ≤ ‖
- Page 141: 133 Definition 15.5. Two vectors y,
- 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
- Page 191 and 192: 183 Plugging these into Bessel’s
134 LESSON 15. LINEAR OPERATORS AND VECTOR SPACES
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