Уравнения математической физики
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МИНИСТЕРСТВО ОБРАЗОВАНИЯ РЕСПУБЛИКИ БЕЛАРУСЬ
Белорусский национальный технический университет
Кафедра «Высшая математика № 3»
УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ
Методические указания для студентов строительных специальностей
Минск
БНТУ
2 0 1 5
ɆɂɇɂɋɌȿɊɋɌȼɈ ɈȻɊȺɁɈȼȺɇɂə ɊȿɋɉɍȻɅɂɄɂ ȻȿɅȺɊɍɋɖ
Ȼɟɥɨɪɭɫɫɤɢɣ нɚɰɢɨɧɚɥɶɧɵɣ тɟɯɧɢɱɟɫɤɢɣ уɧɢɜɟɪɫɢɬɟɬ
Ʉɚɮɟɞɪɚ «ȼɵɫɲɚɹ ɦɚɬɟɦɚɬɢɤɚ ʋ 3»
ɍɊȺȼɇȿɇɂə ɆȺɌȿɆȺɌɂɑȿɋɄɈɃ ɎɂɁɂɄɂ
Ɇɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɫɬɪɨɢɬɟɥɶɧɵɯ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ
Ɇɢɧɫɤ
ȻɇɌɍ
2015
УДК 53:51 (075.8) ББК 22.311я7
М54
Составители:
Н. П. Воронова, А. А. Кузнецова, М. А. Хотомцева, М. Н. Королева
Рецензенты:
В. А. Липницкий, А. В. Метельский
Издание предназначено для студентов строительных специальностей и содержит необходимые теоретические сведения и указания к решению задач. Приведены примеры и варианты заданий.
©Белорусский национальный технический университет, 2015
ȼȼȿȾȿɇɂȿ
ȼ ɫɨɜɪɟɦɟɧɧɨɣ ɧɚɭɤɟ ɢ ɬɟɯɧɢɤɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ, ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɢ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɢɝɪɚɸɬ ɜɫɟ ɛɨɥɶɲɭɸ ɪɨɥɶ. ɗɬɨ ɨɛɭɫɥɨɜɥɟɧɨ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɟɦ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ, ɛɥɚɝɨɞɚɪɹ ɤɨɬɨɪɨɣ ɫɭɳɟɫɬɜɟɧɧɨ ɪɚɫɲɢɪɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɭɫɩɟɲɧɨɝɨ ɩɪɢɦɟɧɟɧɢɹ ɦɚɬɟɦɚɬɢɤɢ ɩɪɢ ɪɟɲɟɧɢɢ ɤɨɧɤɪɟɬɧɵɯ ɡɚɞɚɱ.
Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɤɭɪɫɚ ɫɬɭɞɟɧɬɵ ɞɨɥɠɧɵ ɜɥɚɞɟɬɶ ɨɫɧɨɜɚɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɜ ɨɛɴɟɦɟ ɩɟɪɜɵɯ ɞɜɭɯ ɤɭɪɫɨɜ ɭɧɢɜɟɪɫɢɬɟɬɚ.
ɐɟɥɶɸ ɧɚɫɬɨɹɳɢɯ ɦɟɬɨɞɢɱɟɫɤɢɯ ɭɤɚɡɚɧɢɣ ɹɜɥɹɟɬɫɹ ɨɡɧɚɤɨɦɥɟɧɢɟ ɫ ɨɫɧɨɜɧɵɦɢ ɩɨɧɹɬɢɹɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɨɫɜɨɟɧɢɟ ɦɟɬɨɞɨɜ ɢ ɫɩɨɫɨɛɨɜ ɪɟɲɟɧɢɹ ɨɫɧɨɜɧɵɯ ɡɚɞɚɱ.
Ɂɚɞɚɱɢ ɤɭɪɫɚ ɫɜɨɞɹɬɫɹ ɤ ɢɡɭɱɟɧɢɸ ɨɫɧɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɨɫɜɨɟɧɢɹ ɞɪɭɝɢɯ ɩɪɢɤɥɚɞɧɵɯ ɞɢɫɰɢɩɥɢɧ,
ɢɪɚɡɜɢɬɢɸ ɩɪɚɤɬɢɱɟɫɤɢɯ ɧɚɜɵɤɨɜ ɪɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɡɚɞɚɱ. Ɉɫɨɛɨɟ ɦɟɫɬɨ ɜ ɨɜɥɚɞɟɧɢɢ ɞɚɧɧɵɦ ɤɭɪɫɨɦ ɨɬɜɨɞɢɬɫɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɟ ɩɨ ɪɟɲɟɧɢɸ ɬɟɤɭɳɢɯ ɢ ɢɧɞɢɜɢɞɭɚɥɶɧɵɯ ɞɨɦɚɲɧɢɯ ɡɚɞɚɧɢɣ.
Ⱥɜɬɨɪɵ ɜɵɪɚɠɚɸɬ ɛɥɚɝɨɞɚɪɧɨɫɬɶ ɞɨɰɟɧɬɭ ɤɚɮɟɞɪɵ «ȼɵɫɲɚɹ ɦɚɬɟɦɚɬɢɤɚ ʋ3» ȻɇɌɍ Ʉɪɭɲɟɜɫɤɨɦɭ ȿ.Ⱥ. ɡɚ ɩɨɦɨɳɶ ɜ ɨɛɫɭɠɞɟɧɢɢ
ɢɨɮɨɪɦɥɟɧɢɢ ɞɚɧɧɨɣ ɪɚɛɨɬɵ.
3
Ʌɟɤɰɢɹ ʋ1
1.1. ȼȼȿȾȿɇɂȿ ȼ ɌȿɈɊɂɘ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ ɋ ɑȺɋɌɇɕɆɂ ɉɊɈɂɁȼɈȾɇɕɆɂ
Ȼɨɥɶɲɢɧɫɬɜɨ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɜ ɬɚɤɢɯ ɨɛɥɚɫɬɹɯ, ɤɚɤ ɞɢɧɚɦɢɤɚ ɠɢɞɤɨɫɬɢ, ɷɥɟɤɬɪɢɱɟɫɬɜɨ ɢ ɦɚɝɧɟɬɢɡɦ, ɦɟɯɚɧɢɤɚ, ɨɩɬɢɤɚ, ɬɟɩɥɨɩɟɪɟɞɚɱɚ, ɦɨɝɭɬ ɛɵɬɶ ɨɩɢɫɚɧɵ ɫ ɩɨɦɨɳɶɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ (Ⱦɍɑɉ). Ȼɨɥɶɲɢɧɫɬɜɨ ɭɪɚɜɧɟɧɢɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ – ɷɬɨ Ⱦɍɑɉ, ɯɨɬɹ ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɞɚɥɟɤɨ ɧɟ ɜɫɟ ɜ ɦɚɬɮɢɡɢɤɟ ɢɫɱɟɪɩɵɜɚɟɬɫɹ ɢɦɢ. Ɇɧɨɝɢɟ ɩɪɨɰɟɫɫɵ ɞɥɹ ɫɜɨɟɝɨ ɨɩɢɫɚɧɢɹ ɬɪɟɛɭɸɬ ɩɪɢɫɭɬɫɬɜɢɹ ɢɧɬɟɝɪɚɥɶɧɵɯ ɱɥɟɧɨɜ, ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɢɧɬɟɝɪɚɥɶɧɵɦ ɥɢɛɨ ɞɚɠɟ ɤ ɢɧɬɟɝɪɨ-ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ. Ɉɞɧɚɤɨ ɬɚɤɢɟ ɭɪɚɜɧɟɧɢɹ ɜɵɯɨɞɹɬ ɡɚ ɪɚɦɤɢ ɧɚɫɬɨɹɳɢɯ ɦɟɬɨɞɢɱɟɫɤɢɯ ɭɤɚɡɚɧɢɣ.
ȼ ɨɬɥɢɱɢɟ ɨɬ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (ɈȾɍ), ɜ ɤɨɬɨɪɵɯ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɜ Ⱦɍɑɉ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ (ɧɚɩɪɢɦɟɪ, ɬɟɦɩɟɪɚɬɭɪɚ u(x,t) ɡɚɜɢɫɢɬ ɨɬ ɤɨɨɪɞɢɧɚɬɵ ɯ ɢ ɜɪɟɦɟɧɢ t). Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɡɚɩɢɫɢ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɟ
ɨɛɨɡɧɚɱɟɧɢɹ u |
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Ʉ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɦ Ⱦɍɑɉ ɨɬɧɨɫɹɬɫɹ: |
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ux x uy y uz z |
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– ɬɪɟɯɦɟɪɧɨɟ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ, |
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ȼɨ ɜɫɟɯ ɩɪɢɜɟɞɟɧɧɵɯ ɩɪɢɦɟɪɚɯ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɢ ɡɚɜɢɫɢɬ |
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ɛɨɥɟɟ ɱɟɦ ɨɬ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ. Ɍɚɤɚɹ ɩɟɪɟɦɟɧɧɚɹ ɢ (ɤɨɬɨɪɭɸ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦ) ɧɚɡɵɜɚɟɬɫɹ ɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ ɢɥɢ ɩɪɨɫɬɨ – ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɟɣ. ɉɟɪɟɦɟɧɧɵɟ, ɩɨ ɤɨɬɨɪɵɦ ɩɪɨɢɫɯɨɞɢɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ, ɧɚɡɵɜɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ. ɇɚɩɪɢɦɟɪ, ɜ ɭɪɚɜɧɟɧɢɢ ut ux x ɡɚɜɢɫɢɦɚɹ ɩɟɪɟɦɟɧɧɚɹ (ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ)
4
ɢ(ɯ, t) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ x ɢ t.
Ɇɵ ɭɠɟ ɭɩɨɦɢɧɚɥɢ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɮɢɡɢɱɟɫɤɢɯ ɡɚɤɨɧɨɜ ɩɪɢɪɨɞɵ ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɧɚ ɹɡɵɤɟ Ⱦɍɑɉ. ɉɪɢɦɟɪɨɦ ɦɨɝɭɬ ɛɵɬɶ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ, ɡɚɤɨɧ ɬɟɩɥɨɨɛɦɟɧɚ ɇɶɸɬɨɧɚ, ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɇɶɸɬɨɧɚ, ɭɪɚɜɧɟɧɢɟ ɒɪɺɞɢɧɝɟɪɚ ɜ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɟ. ȼɨ ɜɫɟɯ ɷɬɢɯ ɭɪɚɜɧɟɧɢɹɯ ɮɢɡɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ ɨɩɢɫɵɜɚɸɬɫɹ ɧɚ ɹɡɵɤɟ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɢ ɜɪɟɦɟɧɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. ɉɪɨɢɡɜɨɞɧɵɟ ɩɨ-
ɹɜɥɹɸɬɫɹ ɜ ɭɪɚɜɧɟɧɢɹɯ ɩɨɬɨɦɭ, ɱɬɨ ɨɧɢ ɨɩɢɫɵɜɚɸɬ ɜɚɠɧɟɣɲɢɟ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ (ɬɚɤɢɟ, ɤɚɤ ɫɤɨɪɨɫɬɶ, ɭɫɤɨɪɟɧɢɟ, ɫɢɥɚ, ɬɪɟɧɢɟ, ɩɨɬɨɤ, ɬɨɤ ɢ ɬ. ɞ.). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɨɡɧɢɤɚɸɬ Ⱦɍɑɉ, ɫɨɞɟɪɠɚɳɢɟ ɧɟɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ, ɤɨɬɨɪɭɸ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ (ɬ.ɟ. ɨɩɪɟɞɟɥɢɬɶ ɟɟ ɡɧɚɱɟɧɢɹ ɞɥɹ ɥɸɛɵɯ ɡɧɚɱɟɧɢɣ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ).
Ɍɢɩɵ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ
ɍɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɦɨɠɧɨ ɤɥɚɫɫɢɮɢɰɢɪɨɜɚɬɶ ɩɨ ɦɧɨɝɢɦ ɩɪɢɡɧɚɤɚɦ. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɭɪɚɜɧɟɧɢɣ ɜɚɠɧɚ ɩɨɬɨɦɭ, ɱɬɨ ɞɥɹ ɤɚɠɞɨɝɨ ɤɥɚɫɫɚ ɫɭɳɟɫɬɜɭɟɬ ɫɜɨɹ ɨɛɳɚɹ ɬɟɨɪɢɹ ɢ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ.
ɉɪɢɜɟɞɟɦ ɲɟɫɬɶ ɨɫɧɨɜɧɵɯ ɦɟɬɨɞɨɜ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɭɪɚɜɧɟɧɢɣ.
1. ɉɨɪɹɞɨɤ ɭɪɚɜɧɟɧɢɹ. ɉɨɪɹɞɤɨɦ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɚɢɜɵɫɲɢɣ ɩɨɪɹɞɨɤ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɜɯɨɞɹɳɢɯ ɜ ɭɪɚɜɧɟɧɢɟ.
ut |
ux x |
– ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, |
ut |
u ux x x sin x |
– ɭɪɚɜɧɟɧɢɟ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ. |
2. ɑɢɫɥɨ ɩɟɪɟɦɟɧɧɵɯ. ɑɢɫɥɨɦ ɩɟɪɟɦɟɧɧɵɯ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ. ɇɚɩɪɢɦɟɪ,
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3. Ʌɢɧɟɣɧɨɫɬɶ. ɍɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɛɵɜɚɸɬ ɥɢɧɟɣɧɵɦɢ ɢ ɧɟɥɢɧɟɣɧɵɦɢ. ȼ ɥɢɧɟɣɧɵɟ ɭɪɚɜɧɟɧɢɹ ɡɚɜɢɫɢɦɚɹ ɩɟɪɟɦɟɧɧɚɹ (ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ) ɢ ɜɫɟ ɟɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɜɯɨɞɹɬ ɥɢɧɟɣɧɵɦ ɨɛɪɚɡɨɦ (ɬ.ɟ. ɧɟ ɭɦɧɨɠɚɸɬɫɹ ɞɪɭɝ ɧɚ ɞɪɭɝɚ, ɧɟ ɜɨɡɜɨɞɹɬɫɹ ɜ ɤɜɚɞ-
ɪɚɬ ɢ ɬ. ɞ.). ɇɚɩɪɢɦɟɪ, ɥɢɧɟɣɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɞɜɭɦɹɧɟɡɚɜɢɫɢɦɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɜɢɞɚ:
5
Aux x Bux y Cu y y Dux Eu y Fu G |
(1.1) |
ɝɞɟ Ⱥ, ȼ, ɋ, D, E, F ɢ G – ɤɨɧɫɬɚɧɬɵ ɢɥɢ ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ ɯ ɢ ɭ. ɇɚɩɪɢɦɟɪ,
u |
t t |
e tu |
x x |
sint |
– ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ, |
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u ux x x sin x |
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4.Ɉɞɧɨɪɨɞɧɨɫɬɶ. ɍɪɚɜɧɟɧɢɟ (1.1) ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɟɫɥɢ ɩɪɚ-
ɜɚɹ ɱɚɫɬɶ G(ɯ,ɭ) ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɚ ɧɭɥɸ ɞɥɹ ɜɫɟɯ ɯɢ ɭ. ȿɫɥɢ G(ɯ,ɭ) ɧɟ ɪɚɜɧɚ ɬɨɠɞɟɫɬɜɟɧɧɨ ɧɭɥɸ, ɬɨɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹɧɟɨɞɧɨɪɨɞɧɵɦ.
5.ȼɢɞɵ ɤɨɷɮɮɢɰɢɟɧɬɨɜ. ȿɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬɵ Ⱥ, ȼ, ɋ, D, ȿ ɢ F ɭɪɚɜɧɟɧɢɹ (1.1) ɩɨɫɬɨɹɧɧɵ, ɬɨ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ (ɢɧɚɱɟ ɫɩɟɪɟɦɟɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ).
6.Ɍɪɢ ɨɫɧɨɜɧɵɯ ɬɢɩɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ. ȼɫɟ ɥɢɧɟɣɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɜɢɞɚ (1.1) ɨɬɧɨɫɹɬɫɹ ɤ ɨɞɧɨɦɭ ɢɡɬɪɟɯɬɢɩɨɜ: ɚ) ɩɚɪɚɛɨɥɢɱɟɫɤɢɣ, ɛ) ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɣ, ɜ) ɷɥɥɢɩɬɢɱɟɫɤɢɣ.
ɉɚɪɚɛɨɥɢɱɟɫɤɢɣ ɬɢɩ. ɍɪɚɜɧɟɧɢɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ ɨɩɢɫɵ-
ɜɚɸɬ ɩɪɨɰɟɫɫɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɞɢɮɮɭɡɢɢ ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɫɥɨɜɢɟɦ B2 4AC 0 .
Ƚɢɩɟɪɛɨɥɢɱɟɫɤɢɣ ɬɢɩ. ɍɪɚɜɧɟɧɢɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ ɨɩɢɫɵ-
ɜɚɸɬ ɤɨɥɟɛɚɬɟɥɶɧɵɟ ɫɢɫɬɟɦɵ ɢ ɜɨɥɧɨɜɵɟ ɞɜɢɠɟɧɢɹ ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɫɥɨɜɢɟɦ B2 4AC ! 0
ɗɥɥɢɩɬɢɱɟɫɤɢɣ ɬɢɩ. ɍɪɚɜɧɟɧɢɹ ɷɥɥɢɩɬɢɱɟɫɤɨɝɨ ɬɢɩɚ ɨɩɢɫɵɜɚɸɬ ɭɫɬɚɧɨɜɢɜɲɢɟɫɹ (ɫɬɚɰɢɨɧɚɪɧɵɟ) ɩɪɨɰɟɫɫɵ ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɫɥɨɜɢ-
ɟɦ B2 4AC 0 .
ɉɊɂɆȿɊɕ:
ɚ) |
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0 , ɩɚɪɚɛɨɥɢɱɟɫɤɢɣ ɬɢɩ, |
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4 , ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɣ ɬɢɩ, |
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ɜ) |
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4 , ɷɥɥɢɩɬɢɱɟɫɤɢɣ ɬɢɩ, |
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ɝ) |
yu |
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yy |
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B2 4AC |
4y , ɬ.ɧ. ɫɦɟɲɚɧɧɵɣ ɬɢɩ, ɷɥ- |
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ɥɢɩɬɢɱɟɫɤɢɣ ɬɢɩ ɩɪɢ y > 0, ɩɚɪɚɛɨɥɢɱɟɫɤɢɣ ɬɢɩ ɩɪɢ y = 0, ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɣ ɬɢɩ ɩɪɢ y < 0 .
6
ɂɡ ɷɬɢɯ ɩɪɢɦɟɪɨɜ ɜɢɞɧɨ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɩɟɪɟɦɟɧɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɢɩ ɭɪɚɜɧɟɧɢɹ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɨɬ ɬɨɱɤɢ ɤ ɬɨɱɤɟ. Ɍɢɩ ɭɪɚɜɧɟɧɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɜɬɨɪɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɢ ɧɢɤɚɤɢɦ ɨɛɪɚɡɨɦ ɧɟ ɫɜɹɡɚɧ ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɩɪɢ ɩɟɪɜɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɫ ɫɚɦɨɣ ɮɭɧɤɰɢɟɣ ɢ ɫɜɨɛɨɞɧɵɦ ɱɥɟɧɨɦ.
ɁȺȾȺɑȺ
ɉɪɨɜɟɞɢɬɟɤɥɚɫɫɢɮɢɤɚɰɢɸɫɥɟɞɭɸɳɢɯɭɪɚɜɧɟɧɢɣɩɨɜɫɟɦɩɪɢɡɧɚɤɚɦ:
a) u |
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1.2. ɄɅȺɋɋɂɎɂɄȺɐɂə Ⱦɍɑɉ. ɉɊɂȼȿȾȿɇɂȿ Ʉ ɄȺɇɈɇɂɑȿɋɄɈɆɍ ȼɂȾɍ
Ɇɵ ɭɠɟ ɭɤɚɡɚɥɢ, ɱɬɨ ɥɢɧɟɣɧɨɟ Ⱦɍɑɉ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɞɜɭɦɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ
Aux x Bux y Cuy y Dux Euy Fu G
(A, B, C, D, E, F, G ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ x ɢ y ɢɥɢ ɦɨɝɭɬ ɛɵɬɶ ɤɨɧɫɬɚɧɬɚɦɢ) ɨɬɧɨɫɢɬɫɹ ɤ ɨɞɧɨɦɭ ɢɡ ɫɥɟɞɭɸɳɢɯ ɬɢɩɨɜ ɜ ɡɚɜɢɫɢɦɨɫɬɢ
ɨɬ ɡɧɚɤɚ ɜɵɪɚɠɟɧɢɹ B2 4AC .
ɋ ɩɨɦɨɳɶɸ ɧɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ [ M(x, y), K \(x, y) , ɤɨɬɨɪɵɟ
ɜɜɨɞɢɦ ɜɦɟɫɬɨ x ɢ y, ɩɪɟɨɛɪɚɡɭɟɦ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤ ɤɚɧɨɧɢɱɟ-
ɫɤɨɦɭ ɜɢɞɭ. Ⱦɥɹ ɷɬɨɝɨ ɫɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɞɢɮɮɟɪɟɧɰɢ-
ɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ A(dy)2 B dy dx C(dx)2 0 , ɤɨɬɨɪɨɟ ɪɚɫɩɚɞɚɟɬɫɹ
ɧɚ ɩɚɪɭ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ 1-ɝɨ ɩɨɪɹɞɤɚ (ɜ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɦ ɢ ɷɥɥɢɩɬɢɱɟɫɤɨɦ ɫɥɭɱɚɟ), ɥɢɛɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɨɞɧɨ ɭɪɚɜɧɟɧɢɟ (ɩɚɪɚɛɨɥɢɱɟɫɤɢɣ ɫɥɭɱɚɣ). Ɉɛɳɢɟ ɢɧɬɟɝɪɚɥɵ
ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ M(x, y) C1 , \(x, y) C2 ɧɚɡɵɜɚɸɬ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ (ɜ ɩɚɪɚɛɨɥɢɱɟɫɤɨɦ ɫɥɭɱɚɟ ɛɭɞɟɬ ɬɨɥɶɤɨ ɨɞɧɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ).
1. Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ ɡɚɦɟɧɚ ɩɟɪɟɦɟɧɧɵɯ (ɨɛɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ - ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɮɭɧɤɰɢɢ!) [ M(x, y) , K \(x, y)
ɩɪɢɜɟɞɟɬ ɭɪɚɜɧɟɧɢɟ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ u[K )([,K,u,u[,uK) (ɢɥɢ
u[[ |
u\\ <([,K,u,u[,uK) , |
ɟɫɥɢ |
ɞɨɩɨɥɧɢɬɟɥɶɧɨ |
ɩɨɥɨɠɢɬɶ |
[ |
0,5(M(x, y) \(x, y)) , K |
0,5(M(x, y) \(x, y)) (ɞɥɹ ɝɢɩɟɪɛɨɥɢɱɟ- |
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7
ɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɟɫɬɶ ɞɜɚ ɤɚɧɨɧɢɱɟɫɤɢɯ ɜɢɞɚ);
2. |
Ⱦɥɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɡɚɦɟɧɚ ɩɟɪɟɦɟɧɧɵɯ [ M(x, y) , |
K |
\(x, y) , ɝɞɟ M(x, y) - ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ, ɚ \(x, y) - ɩɪɨɢɡɜɨɥɶɧɚɹ |
ɮɭɧɤɰɢɹ, ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɚɹ ɫ M(x, y) , ɩɪɢɜɟɞɟɬ ɭɪɚɜɧɟɧɢɟ ɤ ɜɢɞɭ uKK )([,K,u,u[,uK) (ɤɚɧɨɧɢɱɟɫɤɢɣ ɜɢɞ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ);
3. ȼ ɷɥɥɢɩɬɢɱɟɫɤɨɦ ɫɥɭɱɚɟ ɨɛɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ M(x, y) ɢ \(x, y) ɹɜɥɹ-
ɸɬɫɹ ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ, ɩɨɷɬɨɦɭ ɜ ɤɚɱɟɫɬɜɟ ɡɚɦɟɧɵ ɩɟɪɟɦɟɧɧɵɯ ɩɪɢɦɟɧɹɸɬ ɢɯ ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɢ ɦɧɢɦɭɸ ɱɚɫɬɶ
[ |
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ɩɪɢ ɷɬɨɦ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ u[[ |
uKK )([,K,u,u[,uK) |
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ɫɤɢɣ ɜɢɞɷɥɥɢɩɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ).
Ɍɟɩɟɪɶ ɦɵ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ ɪɚɫɫɦɨɬɪɢɦ ɜɨɩɪɨɫ ɩɪɢɜɟɞɟɧɢɹ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ɭɪɚɜɧɟɧɢɟ ɨɛɳɟɝɨ ɜɢɞɚ ɜ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɦ ɫɥɭɱɚɟ
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Aux x Bux y Cu y y Dux Eu y Fu |
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ɒȺȽ 1. ȼɜɟɞɟɦ ɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ [ |
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[(x, y) , K |
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K(x, y) ɬɚɤ, ɱɬɨ- |
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ɛɵ ɭɪɚɜɧɟɧɢɟ ɢɦɟɥɨ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɭɸ ɮɨɪɦɭ. Ⱦɥɹ ɷɬɨɝɨ ɫɧɚɱɚɥɚ |
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ɜɵɱɢɫɥɹɟɦ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ |
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ɒȺȽ 2. ɉɨɞɫɬɚɜɢɦ ɷɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɜ ɭɪɚɜɧɟɧɢɟ (1.2) ɢ ɩɨɫɥɟ ɞɨɜɨɥɶɧɨ ɝɪɨɦɨɡɞɤɢɯ ɜɵɱɢɫɥɟɧɢɣ ɢ ɩɪɢɜɟɞɟɧɢɹ ɩɨɞɨɛɧɵɯ ɩɨɥɭɱɚɟɦ
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ɒȺȽ 3. Ɂɚɞɚɞɢɦɫɹ ɰɟɥɶɸ ɜɵɛɪɚɬɶ [ |
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ɬɚɤɢɦ ɨɛ- |
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ɪɚɡɨɦ, ɱɬɨɛɵ ɤɨɷɮɮɢɰɢɟɧɬɵ |
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ɨɛɪɚɬɢɥɢɫɶ ɜ ɧɭɥɶ. ɗɬɨ ɩɨɡɜɨ- |
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ɥɢɬ ɧɚɦ ɩɪɢɜɟɫɬɢ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ. |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ: |
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ɂɡ ɧɢɯ ɩɨɥɭɱɚɟɦ ɩɚɪɭ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ:
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ɇɚɲɚ ɡɚɞɚɱɚ ɫɜɟɥɚɫɶ ɤ ɧɚɯɨɠɞɟɧɢɸ ɞɜɭɯ ɮɭɧɤɰɢɣ [(x, y) ɢ K(x, y) ɬɚɤɢɯ, ɱɬɨɛɵ ɨɬɧɨɲɟɧɢɹ [x /[y ɢ Kx /Ky ɭɞɨɜɥɟɬɜɨɪɹɥɢ ɭɤɚɡɚɧɧɨɣ ɜɵɲɟ ɩɚɪɟ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ.
ɉɊɂɆȿɊ: Ⱦɚɧɨ ɭɪɚɜɧɟɧɢɟ uxx 4u yy ux 0 (ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ). ȿɝɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɣ
dy |
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Ɋɟɲɢɦ ɢɯ ɨɬɧɨɫɢɬɟɥɶɧɨ y, |
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2x c1, y 2x c2 . Ɂɚɬɟɦ |
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ɜɵɪɚɡɢɦ |
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ɨɪɞɢɧɚɬɵ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɪɚɜɧɟɧɢɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤ. ɂɯ ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɧɚ ɪɢɫɭɧɤɟ.
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɜɢɞɚ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (ɫɦ. ɜɵɲɟ) ɩɨɞɫɬɚɜɥɹɟɦ ɧɨɜɵɟ ɤɨɨɪɞɢɧɚɬɵ [(x, y) ɢ K(x, y) ɜ ɭɪɚɜɧɟɧɢɟ
Au[[ Bu[K CuKK Du[ EuK Fu G . ȼɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɞɥɹ ɧɟɝɨ ɛɵɥɢ ɨɩɪɟɞɟɥɟɧɵ ɪɚɧɟɟ.
Рɢɫ. 1. Ƚɪɚɮɢɤɢ ɫɟɦɟɣɫɬɜɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
ɉɪɢɜɟɞɟɦ ɬɟɩɟɪɶ ɉɊɂɆȿɊ ɬɨɝɨ, ɤɚɤ «ɪɚɛɨɬɚɟɬ» ɨɛɳɢɣ ɦɟɬɨɞ ɩɪɢɜɟɞɟɧɢɹ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ.
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