From 00f9e1b27a7147a0c295ba1c2f03248382d6ed2a Mon Sep 17 00:00:00 2001 From: me Date: Tue, 13 Jan 2026 21:53:01 +0300 Subject: [PATCH] upd --- notes/notes.typ | 1 + подготовка к кр/main.pdf | 3569 ++++++++++++++++++++++++++++++++++++++ подготовка к кр/main.typ | 606 +++++++ 3 files changed, 4176 insertions(+) create mode 100644 подготовка к кр/main.pdf create mode 100644 подготовка к кр/main.typ diff --git a/notes/notes.typ b/notes/notes.typ index fcaff9d..1f26c0a 100644 --- a/notes/notes.typ +++ b/notes/notes.typ @@ -297,3 +297,4 @@ $ e^x + y dot x + sin y dot x + e^y eq C_2 $ + diff --git a/подготовка к кр/main.pdf b/подготовка к кр/main.pdf new file mode 100644 index 0000000..f3e20ce --- /dev/null +++ b/подготовка к кр/main.pdf @@ -0,0 +1,3569 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 9 + /Kids [219 0 R 221 0 R 223 0 R 225 0 R 227 0 R 229 0 R 231 0 R 233 0 R 235 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 13 0 R + /Count 4 +>> +endobj + +3 0 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0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 47 0 R 46 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 45 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 44 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 41 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 40 0 R 39 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 38 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 36 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 35 0 R 34 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 33 0 R 32 0 R] +endobj + +23 0 obj +[31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 31 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 30 0 R 29 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 27 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 26 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R] +endobj + +24 0 obj +<< + /Type /StructElem + /S /Document + /P 14 0 R + /K [187 0 R 186 0 R 185 0 R 184 0 R 183 0 R 182 0 R 180 0 R 179 0 R 178 0 R 177 0 R 176 0 R 175 0 R 174 0 R 173 0 R 171 0 R 170 0 R 169 0 R 168 0 R 167 0 R 166 0 R 165 0 R 164 0 R 163 0 R 162 0 R 161 0 R 160 0 R 159 0 R 158 0 R 157 0 R 156 0 R 155 0 R 154 0 R 152 0 R 151 0 R 150 0 R 149 0 R 148 0 R 147 0 R 146 0 R 145 0 R 144 0 R 143 0 R 142 0 R 141 0 R 140 0 R 139 0 R 138 0 R 137 0 R 136 0 R 135 0 R 134 0 R 132 0 R 131 0 R 130 0 R 129 0 R 128 0 R 127 0 R 126 0 R 125 0 R 124 0 R 123 0 R 122 0 R 120 0 R 119 0 R 118 0 R 117 0 R 116 0 R 115 0 R 114 0 R 113 0 R 112 0 R 111 0 R 110 0 R 109 0 R 108 0 R 107 0 R 106 0 R 105 0 R 104 0 R 103 0 R 102 0 R 101 0 R 100 0 R 99 0 R 98 0 R 97 0 R 96 0 R 95 0 R 93 0 R 92 0 R 91 0 R 90 0 R 89 0 R 88 0 R 87 0 R 86 0 R 85 0 R 84 0 R 83 0 R 82 0 R 81 0 R 80 0 R 79 0 R 78 0 R 77 0 R 76 0 R 75 0 R 74 0 R 73 0 R 72 0 R 70 0 R 69 0 R 68 0 R 66 0 R 65 0 R 64 0 R 63 0 R 61 0 R 60 0 R 59 0 R 58 0 R 57 0 R 56 0 R 55 0 R 54 0 R 53 0 R 52 0 R 51 0 R 50 0 R 49 0 R 48 0 R 47 0 R 46 0 R 45 0 R 44 0 R 43 0 R 42 0 R 41 0 R 40 0 R 39 0 R 38 0 R 37 0 R 36 0 R 35 0 R 34 0 R 33 0 R 32 0 R 31 0 R 30 0 R 29 0 R 28 0 R 27 0 R 26 0 R 25 0 R] +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211] + /Pg 235 0 R +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171] + /Pg 235 0 R +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [135] + /Pg 235 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134] + /Pg 235 0 R +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [85] + /Pg 235 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84] + /Pg 235 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42] + /Pg 235 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [283] + /Pg 233 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [275 276 277 278 279 280 281 282] + /Pg 233 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [274] + /Pg 233 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273] + /Pg 233 0 R +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [250] + /Pg 233 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249] + /Pg 233 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225] + /Pg 233 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [209] + /Pg 233 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208] + /Pg 233 0 R +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [182] + /Pg 233 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [172 173 174 175 176 177 178 179 180 181] + /Pg 233 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171] + /Pg 233 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [141 142 143 144 145 146 147 148 149] + /Pg 233 0 R +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140] + /Pg 233 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [112] + /Pg 233 0 R +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111] + /Pg 233 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [70] + /Pg 233 0 R +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69] + /Pg 233 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [35] + /Pg 233 0 R +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34] + /Pg 233 0 R +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [17] + /Pg 233 0 R +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16] + /Pg 233 0 R +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T + /K [0] + /Pg 233 0 R +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228] + /Pg 231 0 R +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [194] + /Pg 231 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [186 187 188 189 190 191 192 193] + /Pg 231 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [185] + /Pg 231 0 R +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [177 178 179 180 181 182 183 184] + /Pg 231 0 R +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [176] + /Pg 231 0 R +>> +endobj + +61 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [171 62 0 R 175] + /Pg 231 0 R +>> +endobj + +62 0 obj +<< + /Type /StructElem + /S /Formula + /P 61 0 R + /K [172 173 174] + /Pg 231 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [158 159 160 161 162 163 164 165 166 167 168 169 170] + /Pg 231 0 R +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [156 157] + /Pg 231 0 R +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155] + /Pg 231 0 R +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [129 67 0 R] + /Pg 231 0 R +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /Formula + /P 66 0 R + /K [130 131 132] + /Pg 231 0 R +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [118 119 120 121 122 123 124 125 126 127 128] + /Pg 231 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /Formula + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117] + /Pg 231 0 R +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [71 71 0 R] + /Pg 231 0 R +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /Formula + /P 70 0 R + /K [72 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+#align(center)[=== №1] + +$ +y'' + 3y' - 4y = e^(-4x) + x e^(-x) +$ + +Характеристический многочлен + +$ +lambda^2 + 3 lambda - 4 eq 0 arrow.double underline(lambda_1 eq 1), underline(lambda_2 eq -4) +$ + +$ +y_"общ" eq C_1 dot e^(x) + C_2 dot e^(-4x) +$ + +Так как $-4$ -- корень характеристического многочлена, частное решение + +$ +y_"частн"_1 eq underline(A dot e^(-4x) dot x) \ +y'_"частн"_1 eq (A dot e^(-4x) dot x)' eq A(e^(-4x) - 4x dot e^(-4x)) eq underline(A e^(-4x) (1 - 4x)) \ +y''_"частн"_1 eq (A e^(-4x) (1 - 4x))' eq A (-4(1 - 4x)e^(-4x) -4e^(-4x)) eq 4 A e^(-4x) (-1(1 - 4x) - 1) eq underline(4 A e^(-4x) (4x - 2)) +$ + +Подставим в изначальное уравнение + +$ +underbrace(4 A e^(-4x) (4x - 2), y''_"частн"_1) + 3 underbrace(A e^(-4x) (1 - 4x), y'_"частн"_1) - 4 underbrace(A dot e^(-4x) dot x, y_"частн"_1) eq e^(-4x) +$ + +Тогда получим + +$ +4 A (4x - 2) + 3 A (1 - 4x) - 4 A x eq 1 \ +underline(16 A x) - 8 A + 3 A - underline(12 A x) - underline(4 A x) eq 1 \ +underline(0 A x) - 8 A + 3 A eq 1 eq 1 \ +-5 A eq 1 \ +A eq -1/5 +$ + +Тогда + +$ +y_"частн"_1 eq A e^(-4x) x eq -frac(x e^(-4x), 5) +$ + +Так как $lambda eq -1$ не является корнем характеристического многочлена + +$ +y_"частн"_2 eq underline((B x + C) e^(-x)) \ +y'_"частн"_2 eq B e^(-x) - e^(-x) (B x + C) eq underline(e^(-x) (B - B x - C)) \ +y''_"частн"_2 eq -e^(-x) (B - B x - C) - B e^(-x) eq underline(e^(-x) (B x - 2 B + C)) +$ + +Подставим в изначальное уравнение + +$ +underbrace((B x - 2B + C)e^(-x), y''_"частн"_2) + 3 underbrace((B - B x - C)e^(-x), y'_"частн"_2) - 4 underbrace((B x + C)e^(-x), y_"частн"_2) eq x e^(-x) +$ + +$ +(B x - 2B + C) + 3(B - B x - C) - 4 (B x + C) eq x \ +underline(B x) - 2B + C + 3 B underline(- 3 B x) - 3 C underline(- 4 B x) - 4 C eq x \ +underline(- 6 B x) + underbrace(B - 6 C, eq 0) eq x +$ + +Тогда + +$ +- 6 B x eq x arrow.double -6 B eq 1 arrow.double B eq -1/6 +$ + +$ +B - 6 C eq 0 arrow.double -1/6 - 6 C eq 0 arrow.double C eq -1/36 +$ + +Тогда + +$ +y''_"частн"_2 eq (B x + C) e^(-x) eq (-1/6 x - 1/36) e^(-x) eq -1/6 (x + 1/6) e^(-x) +$ + +Тогда ответ + +$ +y eq y_"общ" + y_"частн"_1 + y_"частн"_2 eq C_1 e^x + C_2 e^(-4x) - 1/5 x e^(-4x) - 1/6 (x + 1/6) e^(-x) +$ + +#align(center)[=== №2] + +$ +y'' + 2 y' - 3y = x^2 e^x +$ + +Характеристический многочлен + +$ +lambda^2 + 2 lambda - 3 eq (lambda - 1)(lambda + 3) arrow.double underline(lambda_1 eq 1), underline(lambda_2 eq -3) +$ + +Тогда общее решение будет выглядеть так + +$ +y_"общ" eq C_1 e^x + C_2 e^(-3x) +$ + +Так как $lambda = 1$ -- корень характеристического многочлена, то частное решение будет выглядеть так + +$ +y_"частн" eq (A x^2 + B x + C) x e^x eq underline((A x^3 + B x^2 + C x) e^x) \ +y'_"частн" eq A(x^3 e^x)' + B(x^2 e^x)' + C(x e^x)' eq \ +eq A(3x^2 e^x + x^3 e^x) + B(2x e^x + x^2 e^x) + C(e^x + x e^x) eq \ +eq underline(3A x^2 e^x + 2B x e^x + C e^x + A x^3 e^x + B x^2 e^x + C x e^x) \ +y''_"частн" eq 3A(x^2 e^x)' + 2B(x e^x)' + C(e^x)' + A(x^3 e^x)' + B(x^2 e^x)' + C(x e^x)' eq \ +eq 3A (2x e^x + x^2 e^x) + 2B (e^x + x e^x) + C(e^x) + A(3x^2 e^x + x^3 e^x) + B(2x e^x + x^2 e^x) + C(e^x + x e^x) eq \ +eq (3 A + B) (2 x e^x + x^2 e^x) + (2 B + C) (e^x + x e^x) + C e^x + A(3x^2 e^x + x^3 e^x) eq \ +eq underline(6A x e^x + 6A x^2 e^x + 2B e^x + 4B x e^x + 2C e^x + A x^3 e^x + B x^2 e^x + C x e^x) +$ + +Подставим в начальное уравнение + +$ +underbrace(6A x e^x + 6A x^2 e^x + 2B e^x + 4B x e^x + 2C e^x + A x^3 e^x + B x^2 e^x + C x e^x, y''_"частн") + \ ++ 2 dot (underbrace(3A x^2 e^x + 2B x e^x + C e^x + A x^3 e^x + B x^2 e^x + C x e^x, y'_"частн")) - \ +- 3 underbrace((A x^3 + B x^2 + C x)e^x, y_"частн") eq \ +eq x^2 e^x \ + +6A x e^x + 12 A x^2 e^x + 2 B e^x + 8B x e^x + 4 C e^x eq x^2 e^x \ + +(12A) x^2 e^x + (6A + 8B) x e^x + (2B + 4C) e^x eq x^2 e^x \ + +cases( + 12A eq 1, + 6A + 8B eq 0, + 2B + 4C eq 0 +) arrow.double cases( + A eq 1/12, + B eq -1/16, + C eq 1/32 +)\ +$ + +Тогда частное решение + +$ +y_"частн" eq (A x^3 + B x^2 + C x)e^x eq (1/12 x^3 - 1/16 x^2 + 1/32 x) e^x +$ + +Запишем итоговое решение + +$ +y eq y_"общ" + y_"частн" eq C_1 e^x + C_2 e^(-3x) + (1/12 x^3 - 1/16 x^2 + 1/32 x) e^x +$ + +#align(center)[=== №3] + +$ +y'' - 4y' + 8y eq e^(2x) + sin 2x +$ + +Характеристический многочлен + +$ +lambda^2 - 4 lambda + 8 eq 0 +$ + +Дискриминант + +$ +cal(D) eq (-4)^2 - 4(1 dot 8) eq 16 - 32 eq -16 lt 0 +$ + +Корни + +$ +cases( + lambda_1 eq frac(4 plus sqrt(-16), 2) eq underline(2 + 2i), + lambda_2 eq frac(4 minus sqrt(-16), 2) eq underline(2 - 2i) +) +$ + +Запишем общее решение + +$ +y_"общ" eq e^(2x) (C_1 cos 2x + C_2 sin 2x) +$ + +Запишем частное решение + +Правая часть в виде $P_n (x) dot e^(alpha x)$ + +Тогда частное решение + +$ +y_"частн"_1 eq A e^(2x) \ +y'_"частн"_1 eq A(e^(2x))' eq 2A e^(2x) \ +y''_"частн"_1 eq 2A(e^(2x))' eq 4A e^(2x) +$ + +Подставим в изначальное уравнение + +$ +underbrace(4A e^(2x), y''_"частн"_1) - 4 (underbrace(2A e^(2x), y'_"частн"_1)) + 8 (underbrace(A e^(2x), y_"частн"_1)) eq e^(2x) \ +4A - 8A + 8A eq 1 arrow.double A eq 1/4 +$ + +Тогда частное решение для первого слагаемого равно + +$ +y_"частн"_1 eq 1/4 e^(2x) +$ + + +$ +y_"частн"_2 eq B cos 2x + C sin 2x \ +y'_"частн"_2 eq -2B sin 2x + 2C cos 2x \ +y''_"частн"_2 eq -4B cos 2x - 4C sin 2x +$ + +Теперь все подставим в изначальное уравнение + +$ +underbrace(-4(B cos 2x + C sin 2x), y''_"частн"_2) - 4(underbrace(-2(B sin 2x - C cos 2x), y'_"частн"_2)) + 8(underbrace(B cos 2x + C sin 2x, y_"частн"_2)) eq sin 2x +$ + +$ +cases( + 4B - 8C eq 0, + 4C + 8B eq 1 +) arrow.double cases( + C eq 1/20, + B eq 1/10 +) +$ + +Тогда частное решение для $sin 2x$ + +$ +y_"частн"_2 eq 1/10 cos 2x + 1/20 sin 2x +$ + +Финальное решение + +$ +y eq y_"общ" + y_"частн"_1 + y_"частн"_2 eq e^(2x) (C_1 cos 2x + C_2 sin 2x) + 1/4 e^(2x) + 1/10 cos 2x + 1/20 sin 2x +$ + +#align(center)[= Метод исключения] + +#align(center)[=== №4] + +$ +cases( + x' eq x - 3y, + y' eq 3x + y +) +$ + +Из первого уравнения + +$ +3y eq x - x' arrow.double y eq frac(x - x', 3) +$ + +$ +y' eq 1/3 (x - x')' eq 1/3 (x' - x'') +$ + +$ +underbrace(1/3(x' - x''), y') = 3x + underbrace(1/3 (x - x'), y) space.quad | dot 3 +$ + +$ +x' - x'' = 9x + x - x' \ +x'' - 2x' + 10 x eq 0 \ +cal(D) eq 4 - 4 dot 10 eq -36 \ +sqrt(cal(D)) eq sqrt(-36) eq 6i \ +x(t) eq underline(e^t (C_1 cos 3t + C_2 sin 3t)) \ +x' eq e^t ((C_1 + 3 C_2) cos 3 t + (C_2 - 3 C_1) sin 3 t) \ +y(t) eq 1/3(x - x') eq underline(e^t (C_1 sin 3t - C_2 cos 3t)) +$ + +#align(center)[=== №5] + +$ +cases( + x' eq 2x + y, + y' eq 4y - x +) +$ + +Из первого уравнения + +$ +y eq x' - 2x \ +y' eq x'' - 2x' +$ + +$ +underbrace(x'' - 2x', y') eq underbrace(4x' - 8x, 4y) - x +$ + +$ +x'' - 6x' + 9x eq 0 \ +underline(x(t) eq e^(3t) (C_1 + C_2 t)) +$ + +$ +x'(t) eq 3 C_1 e^(3t) + C_2 (e^(3t) + 3t e^(3t)) +$ + +$ +y(t) eq x' - 2x eq underline(e^(3t) (C_1 + C_2 (1 + t))) +$ + +#align(center)[= Метод Эйлера] + +#align(center)[=== №6] + +$ +(lambda_1 eq 1, lambda_(2, 3) eq plus.minus i) +$ + +$ +cases( + x' eq 2x + 2z - y, + y' eq x + 2z, + z' eq y - 2x - z +) +$ + +Выпишем матрицу системы + +$ +A eq mat( + 2, -1, 2; + 1, 0, 2; + -2, 1, -1; +) +$ + +Найдем собственные числа и собственные векторы матрицы $A$. Для поиска собственных чисел составим характеристическое уравнение + +$ +det(A - lambda I) eq mat( + 2 - lambda, -1, 2; + 1, 0 - lambda, 2; + -2, 1, -1 - lambda; + delim: "|" +) eq (1 - lambda)(lambda^2 + 1) +$ + +Итак, собственные числа: + +$ +lambda_1 eq 1, lambda_2 eq i, lambda_3 eq -i +$ + +Найдем собственные векторы + +$ +(A - lambda_1 I) X_1 eq OO arrow.double.l.r mat( + 1, -1, 2; + 1, -1, 2; + -2, 1, -2; +) vec(xi_1, xi_2, xi_3) eq vec(0, 0, 0) +$ + +$ +X_1 eq C_1 vec(0, 2, 1) +$ + +$ +(A - lambda_2 I) X_2 eq OO arrow.double.l.r mat( + 2 - i, -1, 2; + 1, -i, 2; + -2, 1, -1 - i; +) vec(eta_1, eta_2, eta_3) eq vec(0, 0, 0) +$ + +$ +X_2 eq C_2 vec(-1 - i, -1 - i, 1) +$ + +$ +(A - lambda_3 I) X_3 eq OO arrow.double.l.r mat( + 2 + i, -1, 2; + 1, i, 2; + -2, 1, -1 + i; +) vec(chi_1, chi_2, chi_3) eq vec(0, 0, 0) +$ + +$ +X_3 eq overline(X_2) eq C_3 vec(-1 + i, -1 + i, 1) +$ + +Запишем ответ + +$ +vec(x, y, z) eq C_1 e^(t) vec(0, 2, 1) + C_2 e^(i t) vec(-1 - i, -1 - i, 1) + C_3 e^(-i t) vec(-1 + i, -1 + i, 1) +$ + + +#align(center)[=== №7] + +$ +(lambda_1 eq 0, lambda_2 eq lambda_3 eq 1) +$ + +$ +cases( + x' eq 2x - y - z, + y' eq 3x - 2y - 3z, + z' eq 2z - x + y +) +$ + +Запишем матрицу системы + +$ +A eq mat( + 2, -1, -1; + 3, -2, -3; + -1, 1, 2; +) +$ + +Найдем собственные числа + +$ +det(A - lambda I) eq mat( + 2 - lambda, -1, -1; + 3, -2 - lambda, -3; + -1, 1, 2 - lambda; + delim: "|" +) eq -lambda(lambda - 1)^2 +$ + +Тогда собственные числа + +$ +lambda_1 eq 0, lambda_2 eq lambda_3 eq 1 +$ + +Найдем собственный вектор для $lambda_1 eq 0$ + +$ +(A - lambda_1 I) X_1 eq OO arrow.double.l.r mat( + 2, -1, -1; + 3, -2, -3; + -1, 1, 2; +) vec(xi_1, xi_2, xi_3) eq vec(0, 0, 0) +$ + +$ +X_1 eq C_1 vec(1, 3, -1) +$ + +Для $lambda = 1$ + +$ +(A - 1 I) eq mat( + 1, -1, -1; + 3, -3, -3; + -1, 1, 1; +) +$ + +Ранг этой матрицы = 1. То есть по факту все три уравнения зависимы. Любое можно выразить через другое. Поэтому берем любое, например, первое + +$ +x - y - z eq 0 arrow.double x = y + z +$ + +Теперь $y, z$ -- свободные переменные. + +Первый вектор + +$ +X_2 eq vec(1, 1, 0) +$ + +Второй вектор + +$ +X_3 eq vec(1, 0, 1) +$ + +Получим итоговое решение + +$ +vec(x, y, z) eq C_1 vec(1, 3, -1) + C_2 e^t vec(1, 1, 0) + C_3 e^t vec(1, 0, 1) +$ + += Метод вариации постоянной + +$ +cases( + x' eq y - 5 cos t, + y' eq 2x + y +) +$ + +Запишем систему в матричной форме + +$ +X eq vec(x, y), space.quad X' eq vec(x', y') +$ + +Тогда + +$ +X' eq A X + F(t), space.quad A eq mat(0, 1; 2, 1), space.quad F(t) eq vec(-5 cos t, 0) +$ + +Решим однородную систему + +$ +det(A - lambda I) eq mat(-lambda, 1; 2, 1 - lambda; delim: "|") eq (lambda - 2)(lambda + 1) eq 0\ +lambda_1 eq 2, space lambda_2 eq -1 +$ + +Находим собственные векторы + +$ +(A - 2 I) X_1 eq 0 arrow.double mat(-2, 1; 2, -1) vec(x, y) eq vec(0, 0) +$ + +$ +X_1 eq C_1 vec(1, 2) +$ + +$ +(A + I) V_2 eq 0 arrow.double mat(1, 1; 2, 2) vec(x, y) eq 0 +$ + +$ +X_2 eq C_2 vec(1, -1) +$ + +Общее решение однородной системы + +$ +X_h (t) eq C_1 e^(2t) vec(1, 2) + C_2 e^(-t) vec(1, -1) +$ + +Теперь ищем частное решение в виде + +$ +u'_1 X_1 + u'_2 X_2 eq F(t) +$ + +$ +u'_1 vec(1, 2) + u'_2 vec(1, -1) eq vec(-5 cos t, 0) +$ + +или + +$ +cases( + u'_1 + u'_2 = -5 cos t, + 2 u'_1 - u'_2 eq 0 +) +$ + +Из второго уравнения системы + +$ +u'_2 eq 2u'_1 +$ + +Подставим в первое + +$ +u'_1 + 2u'_1 eq 3u'_1 eq -5 cos t arrow.double u'_1 eq -5/3 cos t \ +u'_2 eq 2 u'_1 eq -10/3 cos t +$ + +$ +u_1 (t) eq integral -5/3 cos t d t eq -5/3 sin t \ +u_2 (t) eq integral -10/3 cos t d t eq -10/3 sin t +$ + +Частное решение + +$ +X_p (t) eq u_1 X_1 + u_2 X_2 eq (-5/3 sin t) vec(1, 2) + (-10/3 sin t) vec(1, -1) eq vec(-5 sin t, 0) +$ + +Финальное решение + +$ +y eq y_"общ" + y_"частн" eq C_1 e^(2t) vec(1, 2) + C_2 e^(-t) vec(1, -1) + vec(-5 sin t, 0) +$ + +#line(length: 50%) + +$ +cases( + u'_1 e^(2 t) + u'_2 e^(-t) eq 0, + u'_1 (2 e^(2 t)) + u'_2 (-e^(-t)) eq -5 cos t +) +$ +