Математика для менеджеров. Часть I. Учебное пособие
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ȼɵɱɢɫɥɢɬɶ e ln2 x dx . |
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lnx t ɬɨɝɞɚ |
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ɬɨt= ɟɫɥɢ ɯ ɟ ɬɨ t=1. |
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10.7 Ƚɟɨɦɟɬɪɢɱɟɫɤɢɟ ɩɪɢɥɨɠɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ
ɉɥɨɳɚɞɶ ɮɢɝɭɪɵ ɨɝɪɚɧɢɱɟɧɧɨɣ ɤɪɢɜɵɦɢ y=f1(x ɢ y=f2(x), [f1(x f2(x @ɢ ɩɪɹɦɵɦɢɯ ɚɢɯ b ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
S b f2 (x) f1(x)dx.
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ɉɪɢɦɟɪ ȼɵɱɢɫɥɢɬɶ ɩɥɨɳɚɞɶ ɮɢɝɭɪɵ ɨɝɪɚɧɢɱɟɧɧɨɣ ɡɚ ɞɚɧɧɵɦɢ ɥɢɧɢɹɦɢ y=–x2, y=–x–2.
Ɋɟɲɟɧɢɟ ɋɞɟɥɚɟɦ ɱɟɪɬɟɠ ɇɚɣɞɟɦ ɚɛɫɰɢɫɫɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɞɚɧɧɵɯ ɥɢɧɢɣ
–x2=–x–2 ɢɥɢ x2–x–2=0,x1=–1, x2=2.Ɂɧɚɱɢɬ
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x2 x 2 dx |
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83 42 4 13 12 2
-3+1,5 + 4 + 2 = 4,5.
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Ɉɛɴɟɦ ɬɟɥɚ ɩɨɥɭɱɟɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ ɜɨɤɪɭɝ ɨɫɢ Ɉɯ ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
V b f 2 (x)dx .
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Ⱦɥɢɧɚ ɤɪɢɜɨɣ ɡɚɞɚɧɧɨɣ ɭɪɚɜɧɟɧɢɟɦ y = f (x),a x b,
ɜɵɪɚɠɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ b |
1 f (x) 2 dx. |
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10.8 ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɢɧɬɟɝɪɚɥɨɜ ɜ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ
ɉɪɢɦɟɪ Ɉɩɪɟɞɟɥɢɬɶ ɨɛɴɟɦ ɩɪɨɞɭɤɰɢɢ ɩɪɨɢɡɜɟɞɟɧɧɨɣ ɪɚ ɛɨɱɢɦ ɡɚ ɬɪɟɬɢɣ ɱɚɫ ɪɚɛɨɱɟɝɨ ɞɧɹ ɟɫɥɢ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɬɪɭɞɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɭɧɤɰɢɟɣ f(t) = 3/(3t +1) + 4.
Ɋɟɲɟɧɢɟ ȿɫɥɢ ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ I W ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɬɪɭɞɚ ɪɚɛɨɱɟɝɨ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ W ɬɨ ɨɛɴɟɦ ɩɪɨɞɭɤɰɢɢ ɩɪɨɢɡɜɟɞɟɧɧɨɣ ɪɚɛɨɱɢɦ ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɨɬ W1 ɞɨ W2 ɛɭɞɟɬ ɜɵɪɚɠɚɬɶɫɹ ɮɨɪɦɭɥɨɣ
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ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ |
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3 = ln 10 + 12 - ln 7 - |
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-8 = ln 10/7 + 4.
ɉɪɢɦɟɪ Ɉɩɪɟɞɟɥɢɬɶ ɡɚɩɚɫ ɬɨɜɚɪɨɜ ɜ ɦɚɝɚɡɢɧɟ ɨɛɪɚɡɭɟɦɵɣ ɡɚ ɬɪɢ ɞɧɹ ɟɫɥɢ ɩɨɫɬɭɩɥɟɧɢɟ ɬɨɜɚɪɨɜ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɭɧɤɰɢɟɣ f(t) = 2t + 5.
Ɋɟɲɟɧɢɟ ɂɦɟɟɦ
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3 9 15 24. |
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112
ɌȿɆȺ 11 ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə
11.1. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ
ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ
F(x, y, y', y'', …,y(n)) = 0,
ɫɜɹɡɵɜɚɸɳɟɟ ɚɪɝɭɦɟɧɬ ɯ ɮɭɧɤɰɢɸ ɭ ɯ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ n-ɝɨ ɩɨɪɹɞɤɚ
ɉɨɪɹɞɨɤ ɧɚɢɜɵɫɲɟɣ ɩɪɨɢɡɜɨɞɧɨɣ ɜɯɨɞɹɳɟɣ ɜ ɞɢɮɮɟɪɟɧ ɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɩɨɪɹɞɤɨɦ ɷɬɨɝɨ ɞɢɮɮɟɪɟɧɰɢ ɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɇɚɩɪɢɦɟɪ
y - x2y + x3 = 0 - ɭɪɚɜɧɟɧɢɟ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ y + 4y + cos x = 0 - ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ/
Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ n-ɝɨ ɩɨ ɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɹ ɭ ij ɯ ɋ1 ɋ2 ɋn), ɤɨɬɨɪɚɹ ɡɚɜɢ ɫɢɬ ɨɬ ɚɪɝɭɦɟɧɬɚ ɯ ɢ n ɧɟɡɚɜɢɫɢɦɵɯ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ ɋ1 ɋ2 ɋn ɨɛɪɚɳɚɸɳɚɹ ɜɦɟɫɬɟ ɫɨ ɫɜɨɢɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɭ/, ɭ// ɭ(n)ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɬɨɠɞɟɫɬɜɨ.
ɑɚɫɬɧɵɦ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɪɟɲɟɧɢɟ ɤɨ ɬɨɪɨɟ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɟɫɥɢ ɩɪɢɞɚɜɚɬɶ ɩɨɫɬɨɹɧ ɧɵɦ ɋ1 ɋ2 ɋn ɨɩɪɟɞɟɥɟɧɧɵɟ ɱɢɫɥɨɜɵɟ ɡɧɚɱɟɧɢɹ
Ƚɪɚɮɢɤ ɤɚɠɞɨɝɨ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɶ ɧɨɣ ɤɪɢɜɨɣ ɉɨɷɬɨɦɭ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɟɦɟɣ ɫɬɜɨ ɢɧɬɟɝɪɚɥɶɧɵɯ ɤɪɢɜɵɯ ȼ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɷɬɨ ɫɟɦɟɣɫɬɜɨ ɡɚɜɢɫɢɬ ɨɬ ɨɞɧɨɣ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɨɫɬɨɹɧɧɨɣ ɜ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɹ n-ɝɨ ɩɨɪɹɞɤɚ - ɨɬ n ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ
ȼ ɡɚɞɚɱɟ Ʉɨɲɢ ɧɚɱɚɥɶɧɨɣ ɡɚɞɚɱɟ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɥɹ ɭɪɚɜɧɟɧɢɹ n-ɝɨ ɩɨɪɹɞɤɚ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ n ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
y(xo) = yo, y (xo) = yo ,..., y(n-1)(xo) = yo(n-1),
ɩɨ ɤɨɬɨɪɵɦ ɨɩɪɟɞɟɥɹɸɬɫɹ Q ɩɨɫɬɨɹɧɧɵɯ ɫ1 ɫ2,..., cn.
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ -ɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɨɛɳɢɣ ɜɢɞ
F(x, y, y ) = 0,
ɢɥɢ ɜɢɞ ɪɚɡɪɟɲɟɧɧɵɣ ɨɬɧɨɫɢɬɟɥɶɧɨ \ :y = f(x, y).
113
11 ɍɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ y f1(x)f2 (y) ɧɚɡɵ
ɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ ȼ ɬɚ ɤɢɯ ɭɪɚɜɧɟɧɢɹɯ ɜɫɟ ɜɵɪɚɠɟɧɢɟ ɤɨɬɨɪɨɟ ɧɟ ɨɬɧɨɫɢɬɫɹ ɤ ɩɪɨɢɡ ɜɨɞɧɨɣ ɮɭɧɤɰɢɢ y ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ
ɞɜɭɯ ɮɭɧɤɰɢɣ – ɨɞɧɨɣ ɡɚɜɢɫɹɳɟɣ ɬɨɥɶɤɨ ɨɬ ɯ ɚ ɞɪɭɝɨɟ – ɬɨɥɶɤɨ
ɨɬ |
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ɩɪɟɞɫɬɚɜɢɦ y dxdy ɩɨɥɭɱɢɦ |
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ɉɨɫɥɟɞɧɟɟ ɪɚɜɟɧɫɬɜɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɪɚɜɟɧɫɬɜɨ ɞɢɮɮɟɪɟɧɰɢɚɥɨɜ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɨɬ ɨɛɟɢɯ ɱɚɫɬɟɣ ɛɭɞɭɬ ɨɬɥɢɱɧɵ ɥɢɲɶ ɤɨɧɫɬɚɧɬɨɣ
dy
f2 (y) f1(x)dx C.
ɉɪɢɦɟɪ. Ɋɟɲɢɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ y - xy .
Ɋɟɲɟɧɢɟ:
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Ʉ ɭɪɚɜɧɟɧɢɹɦ ɫ ɪɚɡɞɟɥɹɸɳɟɣɫɹ ɩɟɪɟɦɟɧɧɨɣ ɬɚɤɠɟ ɨɬɧɨ ɫɹɬɫɹ ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ
M1(x)N1(y)dx M2 (x)N2 (x)dy 0;
M1(x)N1(y)dx M2 (x)N2 (y)dx;
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ɉɪɢɦɟɪ ɇɚɣɬɢ |
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ɭɪɚɜɧɟɧɢɹ (1 x)ydx (1 y)xdy 0.
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ln x x y ln y C – ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ Ⱦɍ
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɧɚɰɢɨɧɚɥɶɧɵɣ ɞɨɯɨɞ < ɜɨɡɪɚɫɬɚɟɬ ɫɨ ɫɤɨɪɨ ɫɬɶɸ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɟɝɨ ɜɟɥɢɱɢɧɟ dYdt kY ɢ ɩɭɫɬɶ ɤɪɨɦɟ
ɬɨɝɨ ɞɟɮɢɰɢɬ ɜ ɪɚɫɯɨɞɚɯ ɩɪɚɜɢɬɟɥɶɫɬɜɚ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɞɨɯɨɞɭ < ɩɪɢ ɤɨɷɮɮɢɰɢɟɧɬɟ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ q Ⱦɟɮɢɰɢɬ ɜ ɪɚɫɯɨɞɚɯ ɩɪɢɜɨɞɢɬ ɤ ɜɨɡɪɚɫɬɚɧɢɸ ɧɚɰɢɨɧɚɥɶɧɨɝɨ ɞɨɥɝɚ
D:dD/dt = qY.
Ɂɞɟɫɶ ɦɵ ɫɱɢɬɚɟɦ ɩɟɪɟɦɟɧɧɵɟ < ɢ ' ɧɟɩɪɟɪɵɜɧɵɦɢ ɢ ɞɢɮ ɮɟɪɟɧɰɢɪɭɟɦɵɦɢ ɮɭɧɤɰɢɹɦɢ ɜɪɟɦɟɧɢ t ɉɭɫɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨ ɜɢɹ ɢɦɟɸɬ ɜɢɞ< <o ɢ ' 'o ɩɪɢ W ɂɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɦɵ ɩɨɥɭɱɚɟɦ ɭɱɢɬɵɜɚɹɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ < <o e k t. ɉɨɞɫɬɚɜ ɥɹɹ < ɜɨ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɚɟɦ G' GW T<o e k t.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
D = (q/ k) Yo e k t ɋ ɝɞɟ ɋ FRQVW ɤɨɬɨɪɭɸ ɦɵ ɨɩɪɟɞɟɥɢɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɉɨɞɫɬɚɜɥɹɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɜ ɩɨɥɭɱɟɧ ɧɨɟ ɪɟɲɟɧɢɟ ɦɵ ɩɨɥɭɱɚɟɦ 'o = (q/ k)Yo ɋ ɂɬɚɤ ɨɤɨɧɱɚɬɟɥɶ ɧɨ
D = Do+(q/ k)Yo (e k t -1),
ɬɨ ɟɫɬɶ ɧɚɰɢɨɧɚɥɶɧɵɣ ɞɨɥɝ ɜɨɡɪɚɫɬɚɟɬ ɫ ɬɨɣ ɠɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ k ɱɬɨ ɢ ɧɚɰɢɨɧɚɥɶɧɵɣ ɞɨɯɨɞ
11 Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ y'+ȡ(x)y=f(x) ɝɞɟ ȡ(x ɢf (x) ɧɟɩɪɟ ɪɵɜɧɵɟ ɮɭɧɤɰɢɢ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
115
ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ y'+3y=e2x ɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɯ 0,
ɭ 1.
Ɋɟɲɟɧɢɟ Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ Ɂɞɟɫɶ ȡ(x)= ɢf (x)=e2x.
Ɋɟɲɟɧɢɟ ɢɳɟɦ ɜ ɜɢɞɟ y=UÂȣ ɝɞɟ U ɢ ȣ – ɧɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ ɨɬ ɯ ɇɚɯɨɞɢɦ y'=U'ȣ Uȣ' ɢ ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɭɪɚɜɧɟɧɢɟ
ɡɧɚɱɟɧɢɟ y ɢ y', ɩɨɥɭɱɚɟɦ U ȣ Uȣ Uȣ e2xɢɥɢ U ȣ U ȣ 3ȣ e2x.
ɇɚɣɞɟɦ ɨɞɧɨ ɡɧɚɱɟɧɢɟ ȣ ɩɪɢ ɤɨɬɨɪɨɦ ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛ ɤɚɯ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ȣ ȣ . ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹ ɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ Ɋɟɲɚɹ ɟɝɨ ɩɨɥɭɱɚɟɦ
d |
3 0, |
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3dx,ln ȣ =–3x, ȣ |
–3x |
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ɉɨɞɫɬɚɜɥɹɟɦ ɧɚɣɞɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ȣ ɜ ɢɫɯɨɞɧɨɟ ɞɢɮɮɟ ɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ
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ɂɬɚɤ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
y1e5x C e 3x .
5
ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ Ⱦɥɹ ɷɬɨɝɨ ɩɨɞɫɬɚɜɢɦ ɧɚɱɚɥɶ ɧɵɟ ɭɫɥɨɜɢɹ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɢ ɧɚɣɞɟɦ ɋ
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ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ y |
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11 Ʌɢɧɟɣɧɵɟ ɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ y''+ȡy'+qy=f(x) ɝɞɟ ɢ q – ɜɟɳɟɫɬɜɟɧ ɧɵɟ ɱɢɫɥɚ f (x) – ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɵɦ
116
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧ ɬɚɦɢ
Ɋɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɜɢɞɚ y''+ȡy'+qy=0,
ɭ ɤɨɬɨɪɨɝɨ ɩɪɚɜɚɹ ɱɚɫɬɶ f (x) ɪɚɜɧɚ ɧɭɥɸ Ɍɚɤɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵ ɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ
ɍɪɚɜɧɟɧɢɟ
K2+ȡ.+q=0
ɧɚɡɵɜɚɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɟɦ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɤɜɚɞɪɚɬɧɵɦ
ɭɪɚɜɧɟɧɢɟɦ ɢɦɟɸɳɢɦ ɞɜɚ ɤɨɪɧɹ Ɉɛɨɡɧɚɱɢɦ ɢɯ ɱɟɪɟɡ Ʉ1 ɢ Ʉ2. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ ɡɚɜɢ
ɫɢɦɨɫɬɢ ɨɬ ɜɟɥɢɱɢɧɵ ɞɢɫɤɪɢɦɢɧɚɧɬɚD=ȡ2–4qɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭ ɸɳɢɦ ɨɛɪɚɡɨɦ
ɉɪɢ D>0 ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɟɳɟ ɫɬɜɟɧɧɵɟ ɢ ɪɚɡɥɢɱɧɵɟ Ʉ1 Ʉ2 ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
y C1eK1x C2eK2x .
ɉɪɢ D= ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɟɳɟ ɫɬɜɟɧɧɵɟ ɢ ɪɚɜɧɵɟ Ʉ1 Ʉ2 Ʉ ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ y eKx (C1 C2x).
ȿɫɥɢ D<0 ɬɨ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɤɨɦɩɥɟɤɫɧɵɟ |
K1,2 |
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y = eĮx (C1cosȕx+C2sinȕx). |
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ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɭɪɚɜɧɟɧɢɟ y''+y'–2y=0. |
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Ɋɟɲɟɧɢɟ |
ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ |
ɜɢɞ |
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K2+K–2=0 ɟɝɨ ɤɨɪɧɢ Ʉ1 Ʉ2 |
= – ɜɟɳɟɫɬɜɟɧɧɵɟ ɢ ɪɚɡɥɢɱ |
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ɧɵɟ Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ y = C1ex +C2e–2x. |
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ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ y''–2y'+y=0. |
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Ɋɟɲɟɧɢɟ ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ Ʉ2– |
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ɟɝɨ ɤɨɪɧɢ Ʉ1 |
Ʉ2 = 1 – ɜɟɳɟɫɬɜɟɧɧɵɟ ɢ ɪɚɜɧɵɟ Ɉɛ |
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ɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ y = ex (C1+C2x).
117
ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹy''– 4y'+13y=0.
Ɋɟɲɟɧɢɟ ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ Ʉ2–Ʉ ɟɝɨ ɤɨɪɧɢ Ʉ1 = 2+3i Ʉ2 = 2–3i ɤɨɦɩɥɟɤɫɧɵɟ Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ y = e2x (C1cos3x+C2sin3x).
11 Ʌɢɧɟɣɧɵɟ ɧɟɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɥɢɧɟɣɧɨɟ ɧɟɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨ ɪɨɝɨɩɨɪɹɞɤɚ
y''+ȡx+qy=f(x),
ɝɞɟ f(x) – ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ ɨɬɥɢɱɧɚɹ ɨɬ ɧɭɥɹ
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɬɚɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
~
ɫɭɦɦɭ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ y ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɨɛɳɟɝɨ
ɪɟɲɟɧɢɹ yɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ:
~. y yo y
ɉɨɫɤɨɥɶɤɭ ɧɚɯɨɠɞɟɧɢɟ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɦɵ ɭɠɟ ɪɚɫɫɦɨɬɪɟɥɢ ɬɨ ɨɫɬɚɸɬɫɹ ɪɚɫɫɦɨɬɪɟɬɶ ɜɨɩɪɨɫ ɨ ɧɚɯɨɠɞɟɧɢɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ Ɋɚɫɫɦɨɬɪɢɦ ɪɚɡɥɢɱɧɵɟ ɜɢɞɵ ɩɪɚ ɜɵɯ ɱɚɫɬɟɣ ɧɟɨɞɧɨɪɨɞɧɨɝɨɭɪɚɜɧɟɧɢɹ
ɉɭɫɬɶ ɩɪɚɜɚɹ ɱɚɫɬɶ ɢɦɟɟɬ ɜɢɞ f(x)=e xP~n(x ɝɞɟ Pn(x) – ɦɧɨɝɨɱɥɟɧ ɫɬɟɩɟɧɢ n Ɍɨɝɞɚ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ y ɢɳɟɦ ɜ ɜɢɞɟ
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ɝɞɟ Qn(x) – ɦɧɨɝɨɱɥɟɧ ɬɨɣ ɠɟ ɫɬɟɩɟɧɢ ɱɬɨ ɢ |
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Pn(x ɚ r – ɱɢɫɥɨ ɩɨɤɚɡɵɜɚɸɳɟɟ ɫɤɨɥɶɤɨ ɪɚɡ Į ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ y''– 2y'+y=x2+1.
Ɋɟɲɟɧɢɟ Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɨɞɧɨɪɨɞɧɨ ɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ yo = ex (C1+C2x) ɫɦ ɩɪɢɦɟɪ ɜɵɲɟ Ɍɚɤ ɤɚɤ ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ ɹɜɥɹɟɬɫɹ ɦɧɨɝɨɱɥɟɧɨɦ ɜɬɨɪɨɣ ɫɬɟ ɩɟɧɢ ɢ ɧɢ ɨɞɢɧ ɢɡ ɤɨɪɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ
x2 2x 1 0 |
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ɧɟ ɪɚɜɟɧ ɧɭɥɸ Ʉ1 Ʉ2=1 ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ |
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Bx C ɝɞɟ Ⱥ ȼ ɋ – ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɷɮ |
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+Bx+C ɢ ɩɨɞɫɬɚɜ |
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ɜ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ |
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ɧɚɯɨɞɢɦ 2A–4Ax–2B+Ax2+Bx+C=x2+1 ɢɥɢ Ax2+(B– 4A)x+2A–2B+C=x2+1.
ɉɪɢɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ ɯ ɜ ɨɛɟɢɯ ɱɚɫɬɹɯ ɪɚɜɟɧɫɬɜɚ ɢɦɟɟɦȺ ȼ- Ⱥ Ⱥ- ȼ ɋ ɇɚɯɨ
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ɂɬɚɤ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ |
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ɢɦɟɟɬ |
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ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɢ ɱɚɫɬɧɨɟ ɪɟ |
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ɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ |
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y y 2y 3e2x , |
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Ɋɟɲɟɧɢɟ Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɨɞɧɨɪɨɞɧɨ |
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ɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ yo = C1ex +C2e–2x ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɞɚɧ ɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɬɨɢɬ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɧɨɝɨɱɥɟɧɚ ɧɭɥɟɜɨɣ ɫɬɟɩɟ ɧɢ ɧɚ ɩɨɤɚɡɚɬɟɥɶɧɭɸ ɮɭɧɤɰɢɸ eĮx ɩɪɢ Į Ɍɚɤ ɤɚɤ ɫɪɟɞɢ ɤɨɪ
ɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɟɬ ɤɨɪɧɟɣ ɪɚɜɧɵɯ ɬɨ |
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ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɳɟɦ ɜ ɜɢɞɟ y =AÂe |
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Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɢ ɩɨɞɫɬɚɜɥɹɹ y ɜ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɚɟɦ |
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4 A e2x 2 A e2x 2Ae2x 3e2x |
ɢ 4 A e2x 3e2x |
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ɉɨɞɫɬɚɜɥɹɹ ɧɚɣɞɟɧɧɨɟ ɡɧɚɱɟɧɢɟ Ⱥ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ y , |
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ɧɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ y |
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ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢ ɹɦ Ⱦɥɹ ɷɬɨɝɨ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɭ y C1ex 2C2e 2x 64e2x .
ɉɨɞɫɬɚɜɥɹɟɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɜ ɭ ɢ ɭ ɧɚɯɨɞɢɦ ɋ1 ɢ
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ɉɨɞɫɬɚɜɥɹɹ ɧɚɣɞɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɋ1 ɢ ɋ2 ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɭ ɧɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
y 23ex 125 e 2x 34e2x .
ɉɭɫɬɶ ɩɪɚɜɚɹ ɱɚɫɬɶ ɢɦɟɟɬ ɜɢɞ
f(x) e x (a cos x b sin x) ɢ Į ȕi Į–ȕi) ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪ ɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ Ɍɨɝɞɚ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ
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(Acos x Bsin x) . |
ɢɳɟɦ ɜ ɜɢɞɟ y |
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ȿɫɥɢ ɠɟ Į ȕi Į–ȕi ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨ ɝɨ ɭɪɚɜɧɟɧɢɹ ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɚɯɨɞɢɦ ɜ ɜɢɞɟ
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x(Acos x Bsin x) . |
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ɉɪɢɦɟɪ. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ y y sin 2x |
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Ɋɟɲɟɧɢɟ Ɂɞɟɫɶ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ Ʉ2 + 1 = 0 ɢɦɟɟɬ ɤɨɪɧɢ Ʉ1=i Ʉ2 = -i ɉɨɷɬɨɦɭ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɨɨɬɜɟɬɫɬɜɭ ɸɳɟɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ y = C1cosx+C2sinx ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɫɬɨɢɬ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɟ ɮɭɧɤɰɢɹ sin 2x, ɬɨ ɟɫɬɶ
a=0, b ȕ Ɍɚɤ ɤɚɤ ȕ ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢ
ɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɨ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɚɞɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ |
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y Acos2x Bsin2x . |
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ɢ ɩɨɞɫɬɚɜɥɹɹ ɟɝɨ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶ |
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Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ y |
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ɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɢɦ |
3Acos2x 3Bsin2x sin2x ɨɬɤɭɞɚ |
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B 3 ɬ ɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ y |
3sin2x ɚ ɨɛɳɟɟ |
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ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ y C cosx C |
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sinx 1sin2x . |
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