From 82eda2c00adebd3d82241fb90cf6086ec8af4eb6 Mon Sep 17 00:00:00 2001 From: fYmious Date: Mon, 15 Dec 2025 20:00:42 +0300 Subject: [PATCH] upd --- labs/lab2/report.pdf | 2920 ++++++++++++++++++++++++++++++++++++++++++ labs/lab2/report.typ | 332 +++++ 2 files changed, 3252 insertions(+) create mode 100644 labs/lab2/report.pdf create mode 100644 labs/lab2/report.typ diff --git a/labs/lab2/report.pdf b/labs/lab2/report.pdf new file mode 100644 index 0000000..eeeaf9d --- /dev/null +++ b/labs/lab2/report.pdf @@ -0,0 +1,2920 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 6 + /Kids [177 0 R 184 0 R 186 0 R 188 0 R 190 0 R 192 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 7 0 R + /Count 5 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 4 0 R + /Title + /Dest 172 0 R +>> +endobj + +4 0 obj +<< + /Parent 2 0 R + /Next 5 0 R + /Prev 3 0 R + /Title + /Dest 173 0 R +>> +endobj + +5 0 obj +<< + /Parent 2 0 R + /Next 6 0 R + /Prev 4 0 R + /Title + /Dest 174 0 R 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justify: true, + leading: 0.52em, +) +#show link: underline +#set page(footer: context { + if counter(page).get().first() > 1 [ + #align(center)[ + #counter(page).display("1") + ] + ] + if counter(page).get().first() == 1 [ + #align(center)[ + Санкт-Петербург \ 2025 + ] + ] +}) + +#set page(header: context { + if counter(page).get().first() == 1 [ + #align(center)[ + Санкт-Петербургский национальный исследовательский университет информационных технологий, механики и оптики + ] + ] +}) + + + +#show raw.where(block: false): box.with( + fill: luma(240), + inset: (x: 3pt, y: 0pt), + outset: (y: 3pt), + radius: 2pt, +) + +#show raw.where(block: true): block.with( + fill: luma(240), + inset: 10pt, + radius: 4pt, +) + +// title + +#for _ in range(5) { linebreak() } + +#align(center)[Расчетно графическая работа №2] + +#for _ in range(15) { linebreak() } + + +#align(right)[Выполнили:] +#align(right)[Левахин Лев] +#align(right)[Останин Андрей] +#align(right)[Дощенников Никита] +#align(right)[Группы: К3221, К3240] +#align(right)[Проверил:] +#align(right)[Владимир Владимирович Беспалов] + +#pagebreak() + +#outline(title: "Содержание") + +#align(center)[=== Цель] + +Решение неоднородного линейного дифференциального уравнения разными способами. + +Методы будут демонстрироваться на задаче Коши: + +$ +y'' - 4y' + 5y = 4 e^(-2x), space y(0) = 0, space y'(0) = 1 +$ + +#align(center)[=== Метод специальной правой части] + +Запишем характеристическое уравнение: + +$ +r^2 - 4r + 5 = 0 +$ + +Найдем его корни. Посчитаем дискриминант: + +$ +cal(D) = b^2 - 4 a c = 16 - 4 dot 5 = -4 lt 0 +$ + +Тогда корни: + +$ +r_1 = frac(4 + sqrt(-4), 2) = 2 + i, space.quad r_2 = frac(4 - sqrt(-4), 2) = 2 - i +$ + +Запишем общее решение для однородного уравнения. Так как корни комплексные, то общее решение соответствует форме: + +$ +y_"общ" = e^(alpha x) dot (C_1 dot cos beta x + C_2 dot sin beta x) +$ + +В нашем случае $alpha = 2, beta = 1$. Тогда: + +$ +y_"общ" = e^(2 x) dot (C_1 dot cos x + C_2 dot sin x) +$ + +Специальный вид правой части соответствует виду: + +$ +f(x) = P_n (x) dot e^(alpha x) +$ + +где $P_0 (x) = 4 space (n = 0), alpha = -2$. + +Так как $alpha = -2$ не является корнем характеристического уравнения, частное решение представим в следующем виде: + +$ +y_"частн" = e^(alpha x) dot Q_0 (x) = e^(-2 x) dot A +$ + +Найдем первую и вторую производные для частного решения: + +$ +y'_"частн" eq (A e^(-2x))' = -2 A e^(-2x) +$ + +$ +y''_"частн" eq (-2 A e^(-2x))' = 4 A e^(-2x) +$ + +Подставим в исходное уравнение: + +$ +4 A e^(-2x) + 8 A e^(-2x) + 5 A e^(-2x) = 4e^(-2x) +$ + +Вынеся $e^(-2x)$ за скобки получим: + +$ +e^(-2x) (4A + 8A + 5A) = e^(-2x)(4) +$ + +Отсюда видно, что + +$ +4A + 8A + 5A = 4 arrow.double 17A = 4 arrow.double A = frac(4, 17) +$ + +Тогда + +$ +y_"частн" = A e^(-2x) = frac(4, 17) e^(-2x) +$ + +Запишем общее решение: + +$ +y = y_"общ" + y_"частн" = e^(2 x) dot (C_1 dot cos x + C_2 dot sin x) + frac(4, 17) e^(-2x) +$ + +Теперь решим задачу Коши. Чтобы определить коэффициенты, подставим $0$ вместо $x$: + +$ +y = C_1 + frac(4, 17) +$ + +по условию $y(0) = 0$, то есть + +$ +C_1 + frac(4, 17) = 0 arrow.double C_1 = -frac(4, 17). +$ + +Найдем производную $y'$: + +$ +y' = C_1 (e^(2x) cos x)' + C_2 (e^(2x) sin x)' + frac(4, 17)(e^(-2x))' = \ += C_1 (2e^(2x) cos x - e^(2x) sin x) + C_2 (2e^(2x) sin x + e^(2x) cos x) - frac(8, 17)e^(-2x) = \ += e^(2x)((2C_1 + C_2) cos x + (2 C_2 - C_1) sin x) - frac(8, 17) e^(-2x) +$ + +По условию $y'(0) = 1$. Подставим $0$ вместо $x$ и получим: + +$ +y'(0) = 2C_1 + C_2 - frac(8, 17) = 1 +$ + +Подставим $C_1 = -frac(4, 17)$: + +$ +-frac(8, 17) + C_2 - frac(8, 17) = 1 arrow.double C_2 = frac(33, 17) +$ + +Подставим найденные коэффициенты в итоговое решение: + +$ +y = e^(2x) (-frac(4, 17) cos x + frac(33, 17) sin x) + frac(4, 17) e^(-2x). +$ + +#align(center)[=== Метод вариации постоянной] + +Напомним вид общего решения для однородного уравнения: + +$ +y_"общ" = e^(2 x) dot (C_1 dot cos x + C_2 dot sin x) +$ + +Раскроем скобки и получим + +$ +y_"общ" = C_1 e^(2 x) cos x + C_2 e^(2 x) sin x +$ + +Сведем все к системе: + +$ +cases( + C'_1 (x) y_1 + C'_2 (x) y_2 = 0, + C'_1 (x) y'_1 + C'_2 (x) y'_2 = f(x) +) +$ + +где $y_1 = e^(2x) cos x, space y_2 = e^(2x) sin x, space f(x) = 4 e^(-2x)$. + +Найдем производные для $y_1$ и $y_2$: + +$ +y_1' = (e^(2x) cos x)' = 2 e^(2x) cos x - e^(2x) sin x +$ + +$ +y_2' = (e^(2x) sin x)' = 2 e^(2x) sin x + e^(2x) cos x +$ + +Тогда система примет следующий вид: + +$ +cases( + C'_1(x) (e^(2x) cos x) + C'_2(x) (e^(2x) sin x) = 0, + C'_1(x) (2 e^(2x) cos x - e^(2x) sin x) + C'_2(x) (2 e^(2x) sin x + e^(2x) cos x) = 4e^(-2x), +) +$ + +Из первого уравнения системы: + +$ +C'_1 = -C'_2 frac(e^(2x) sin x, e^(2x) cos x) = -C'_2 tg x +$ + +Подставим во второе уравнение системы и вынесем $C'_2 e^(2x)$: + +$ +C'_2 e^(2x) (-tg x(2 cos x - sin x) + 2 sin x + cos x) = 4 e^(-2x) +$ + +Упростим + +$ +-tg x(2 cos x - sin x) = -frac(sin x, cos x) (2 cos x - sin x) = -2 sin x + frac(sin^2 x, cos x) +$ + +Тогда + +$ +(-2 sin x + frac(sin^2 x, cos x)) + 2 sin x + cos x = frac(sin^2 x + cos^2 x, cos x) = frac(1, cos x) +$ + +Вернемся в уравнение 2 системы. С учетом упрощения оно примет следующий вид: + +$ +C'_2 e^(2x) dot frac(1, cos x) = 4 e^(-2x) +$ + +Отсюда + +$ +C_2' = 4e^(-4x) cos x +$ + +Найдем $C'_1$ + +$ +C'_1 = -C'_2 tg x = -4e^(-4x) cos x frac(sin x, cos x) = -4e^(-4x) sin x +$ + +Проинтегрируем и получим + +$ +C_1 = -4 integral e^(-4x) sin x d x = frac(4, 17) e^(-4x) (4 sin x + cos x) \ +C_2 = 4 integral e^(-4x) cos x d x = frac(4, 17) e^(-4x) (4 cos x - sin x) +$ + +Подставим в формулу частного решения: + +$ +y_"частн" = C_1 (x) y_1 + C_2 (x) y_2 = \ += frac(4, 17) e^(-4x) (4 sin x + cos x) dot e^(2x) cos x + frac(4, 17) e^(-4x) (4 cos x - sin x) dot e^(2x) sin x = \ += frac(4, 17) e^(-2x) ((4 sin x + cos x ) cos x + (4 cos x - sin x) sin x) = \ += frac(4, 17) e^(-2x) (4 sin x cos x + 4 cos x sin x + cos^2 x - sin^2 x) = \ += frac(4, 17) e^(-2x) (8 sin x cos x + (cos^2 x + sin^2 x)) = \ += frac(4, 17) e^(-2x) (8 sin x cos x + 1) = \ += frac(4, 17) e^(-2x) (cos 2 x + 4 sin 2x) +$ + +Так как $e^(-2x) cos 2 x$ и $e^(-2x) sin 2x$ -- решения однородного уравнения, то выражение $frac(4, 17) e^(-2x) (cos 2 x + 4 sin 2 x)$ -- это частное решение и часть однородного решения. Тогда окончательно получим + +$ +y_"частн" = frac(4, 17) e^(-2x) +$ + +Запишем общее решение: + +$ +y = y_"общ" + y_"частн" = e^(2x) (-frac(4, 17) cos x + frac(33, 17) sin x) + frac(4, 17) e^(-2x). +$ + +Дальнейшее решение задачи Коши идентично описанному в первом методе. + +#align(center)[=== Операционный метод] +#align(center)[=== Метод разложения в ряд] + + +