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Computer Modern", + size: 14pt +) +#set par( + // first-line-indent: ( + // amount: 1.5em, + // all: true + // ), + justify: true, + leading: 0.52em, +) + +#align(center)[=== Определения] + +1. Дифференциальное уравнение + +Обыкновенным дифференциальным уравнением первого порядка называют уравнение вида + +$ +F(x, y, y') eq 0. +$ + +#align(center)[ИЛИ] + +Дифференциальным уравнением называется уравнение, связывающее независимую переменную $x$, искомую функцию $y eq y(x)$ и её производные $y′, y′′, dots , y(n)$. + +#line(length: 100%) + +2. Решение дифференциального уравнения, общее решение + +Функция $phi$ - решение уравнения, если + +$ +phi in C^1 (a, b); \ +F(x, phi(x), phi'(x)) eq.triple 0 " на " (a, b) +$ + +Другими словами, решением уравнения называют гладкую функцию $phi$, определённую на интервале $(a, b)$, подстановка которой вместо $y$ обращает уравнение в тождество на $(a, b)$. + +Общим решением уравнения называют множество всех его решений. + +#align(center)[ИЛИ] + +Функция $y eq phi(x)$ является решением дифференциального уравнения, если её подстановка в уравнение обращает его в тождество. + +#line(length: 100%) + +3. Задача Коши + +Задачей Коши или начальной задачей для нормального уравнения + +$ +y' eq f(x, y) +$ + +называют задачу нахождения его решения, удовлетворяющего начальному условию + +$ +y(x_0) eq y_0. +$ + +Пара чисел $(x_0, y_0)$ при этом называется начальными данными. + +#line(length: 100%) + +4. Уравнение с разделяющимися переменными + +Уравнение в дифференциалах вида + +$ +P(x) d x plus Q(y) d y eq 0 +$ + +называют уравнением с разделёнными переменными. + +Такое название мотивировано тем, что каждое его слагаемое зависит только от одной переменной. + +Уравнение вида + +$ +p_1 (x) q_1 (y) d x plus p_2 (x) q_2 (y) d y eq 0 +$ + +называют уравнением с разделяющимися переменными. + +#align(center)[ИЛИ] + +Если уравнение $Phi(x, y, y') eq 0$ с помощью алгебраических преобразований удается привести к виду + +$ +y' eq g(x) dot h(y) +$ + +или + +$ +M_1 (x) M_2 (y) d x plus N_1 (x) N_2 (y) d y eq 0, +$ + +то оно называется уравнением с разделяющимися переменными. + +#line(length: 100%) + +5. Однородная функция + +Функция $F(x, y)$ называется однородной функцией степени $alpha$, если при всех допустимых $t$, $x$ и $y$ верно равенство + +$ +F(t x, t y) eq t^alpha F(x, y). +$ + +Пример однородных функций: $x plus y plus z$ (первой степени), $x^2 plus 3 x y plus y^2$ (второй степени), $y/x cos x/y$ (нулевой степени), $frac(sqrt(x plus y), x^2 plus y^2)$ (степени $-3/2$). + +#line(length: 100%) + +6. Однородное ДУ первого порядка + +Пусть $P$ и $Q$ - однородные функции одинаковой степени. Тогда уравнение вида + +$ +P(x, y) d x plus Q(x, y) d y eq 0 +$ + +называется однородным уравнением. + +#align(center)[ИЛИ] + +Дифференциальное уравнение первого порядка называется однородным, если его можно привести к виду: + +$ +y' eq f(y/x) +$ + +#line(length: 100%) + +7. Линейное ДУ первого порядка + +Дифференциальное уравнение вида + +$ +y' eq p(x) y plus q(x), +$ + +называется линейным уравнением первого порядка. + +Название линейное мотивировано тем, что оно составлено из многочленов первой степени по отношению к символам $y$ и $y'$. + +#align(center)[ИЛИ] + +Линейным уравнением первого порядка называется уравнение вида + +$ +y' plus p(x) y eq q(x), +$ + +где $p(x), q(x)$ -- заданные функции. + +#line(length: 100%) + +8. Уравнение Бернулли + +Уравнением Бернулли называют уравнение вида + +$ +y' eq p(x) y plus q(x) y^alpha, +$ + +где $alpha in RR without {0, 1}$. + +Разделив данное уравнение на $y^alpha$, находим + +$ +frac(y', y^alpha) eq p(x) y^(1 minus alpha) plus q(x). +$ + +Отсюда видно, что замена $z eq y^(1 minus alpha)$ сводит уравнение к линейному. + +#align(center)[ИЛИ] + +Уравнением Бернулли называется уравнение вида + +$ +y' plus p(x) y eq q(x) y^a, space.quad "где" a eq "const", a eq.not 0, a eq.not 1 +$ + +#line(length: 100%) + +9. Уравнение в полных дифференциалах + +Уравнение + +$ +P(x, y) d x plus Q(x, y) d y eq 0 +$ + +называют уравнением в полных дифференциалах, если существует такая функция $u$, что + +$ +d u eq P(x, y) d x plus Q(x, y) d y, +$ + +то есть $u'_x eq P, u'_y eq Q$. + +#align(center)[ИЛИ] + +Дифференциальное уравнение вида + +$ +M(x, y) d x plus N(x, y) d x eq 0 +$ + +называется уравнением в полных дифференциалах, если его левая часть представляет собой полный дифференциал некоторой функции $u(x, y)$: + +$ +M d x plus N d y eq d u eq frac(partial u, partial x) d x plus frac(partial u, partial y) d y. +$ + +Условие того, что $M d x plus N d y$ представляет собой полный дифференциал: + +$ +frac(partial M, partial y) eq frac(partial N, partial x). +$ + +#line(length: 100%) + +10. Особое решение ДУ + +Решение $y eq phi(x)$ дифференциального уравнения + +$ +Phi(x, y, y') eq 0 +$ + +называется особым, если в каждой его точке нарушается свойство единственности, то есть если через каждую его точку ($x_0, y_0$) кроме этого решения проходит и другое решение, имеющее в точке ($x_0, y_0$) ту же касательную, что и решение $y eq phi(x)$, но не совпадающее с ним в сколь угодно малой окрестности ($x_0, y_0$). График особого решения будем называть особой интегральной кривой уравнения. + +#align(center)[ИЛИ] + +Решение $phi$ на $(a, b)$ уравнения $y' eq f(x, y)$ называется особым, если для любой точки $x_0 in (a, b)$ найдется решение $psi$ того же уравнения, такое что + +$ +phi(x_0) eq psi(x_0) +$ + +при этом $phi eq.triple psi$ в любой сколь угодно малой окрестности точки $x_0$. + +Более кратко это выражают словами: интегральная кривая уравнения $y' eq f(x, y)$ является особой, если в каждой её точке нарушается единственность решения задачи Коши. + +#line(length: 100%) + +11. ДУ высшего порядка, задача Коши для него + +Дифференциальным уравнением $n$-го порядка называют уравнение вида + +$ +F(x, y, y', dots, y^((n))) eq 0. +$ + +Функция $phi$ -- решение уравнения на $(a, b)$, если + +$ +phi in C^n (a, b); \ +F(x, phi(x), phi'(x), dots, phi^((n))(x)) eq.triple 0 " на " (a, b). +$ + +Каноническим уравнением будем называть уравнение + +$ +y^((n)) eq f(x, y, y', dots, y^((n minus 1))), +$ + +разрешённое относительно старшей производной. + +Задачей Коши для канонического уравнения называют задачу нахождения его решения, удовлетворяющего начальным условиям + +$ +y(x_0) eq y_0, y'(x_0) eq y'_0, dots, y^((n minus 1)) (x_0) eq y_0^((n minus 1)). +$ + +Набор чисел $(x_0, y_0, y'_0, dots, y_0^((n minus 1)) )$ при этом называют начальными данными. + +#align(center)[ИЛИ] + +Обыкновенное дифференциальное уравнение $n$-го порядка имеет вид + +$ +Phi(x, y, y', y'', dots, y^((n))) eq 0, +$ + +или в решенном относительно старшей производной $y^((n))$, вид + +$ +y^((n)) eq f(x, y, y', y'', dots, y^((n minus 1))). +$ + +Всякая функция $y(x)$, имеющая непрерывные производные вплоть до $n$-го порядка и удовлетворяющая уравнению, называется решением этого уравнения, а сама задача нахождения решений дифференциального уравнения называется задачей интегрирования дифференциального уравнения. + +#line(length: 100%) + +12. Линейное ДУ $n$-го порядка. Однородное, неоднородное + +Линейным дифференциальным уравнением порядка $n$ называется уравнение вида + +$ +y^((n)) plus p_(n minus 1) (t) y^((n minus 1)) plus dots plus p_1 (t) dot(y) plus p_0 (t) y eq q(t), +$ + +где $p_0, p_1, dots, p_(n minus 1), q in C(a, b)$. + +Если $q eq.triple 0$ на $(a, b)$, то уравнение, то есть + +$ +y^((n)) plus p_(n minus 1) (t) y^((n minus 1)) plus dots plus p_1 (t) dot(y) plus p_0 (t) y eq 0, +$ + +называется однородным, в противном случае -- неоднородным. + +#align(center)[ИЛИ] + +Уравнение вида + +$ +y^((n)) plus a_1 (x) y^((n minus 1)) plus dots plus a_(n minus 1) (x) y' plus a_n (x) y eq 0 +$ + +называется линейным однородным дифференциальным уравнением $n$-го порядка. + +Уравнение вида + +$ +y^((n)) plus a_1 (x) y^((n minus 1)) plus dots plus a_(n minus 1) (x) y' plus a_n (x) y eq f(x) +$ + +называется линейным неоднородным дифференциальным уравнением $n$-го порядка. + +#line(length: 100%) + +13. Линейная независимость функций + +#line(length: 100%) + +14. Определитель Вронского + +Определителем Вронского (или вронскианом) функций $y_1, y_2, dots, y_n in C^((n minus 1)) (a, b)$ называют + +$ +W (t) colon.eq mat( + y_1 (t), y_2 (t), dots, y_n (t); + dot(y)_1 (t), dot(y)_2 (t), dots, dot(y)_n (t); + dots, dots, dots, dots; + y^((n minus 1)) (t), y_2^((n minus 1)), dots, y_n^((n minus 1)) (t); delim: "|") +$ + +#line(length: 100%) + +15. Фундаментальная система решений + +Фундаментальной системой решений системы уравнений называется совокупность её $n$ линейно независимых решений. + +#align(center)[ИЛИ] + +Любой набор из $n$ линейно независимых решений $y_1 (x), y_2 (x), dots, y_n (x)$ уравнения $y^((n)) plus a_1 (x) y^((n minus 1)) plus dots plus a_(n minus 1) (x) y' plus a_n (x) y eq 0$ называется фундаментальной системой решений этого уравнения. + +#line(length: 100%) + +16. Характеристический многочлен + +Многочлен + +$ +p(lambda) colon.eq lambda^n plus a_(n minus 1) lambda^(n minus 1) plus dots plus a_1 lambda plus a_0 +$ + +называется характеристическим многочленом уравнения $y^((n)) plus a_(n minus 1) y^((n minus 1)) plus dots plus a_1 dot(y) plus a_0 y eq f(t)$, а его корни -- характеристическими числами того же уравнения. + +#line(length: 100%) + +17. Система ДУ, решение системы + +Система дифференциальных уравнений -- это набор дифференциальных уравнений, решаемых совместно. Решение системы -- это набор функций, который удовлетворяет всем уравнениям системы. Такая форма записи системы называется нормальной формой Коши: + +$ +cases(frac(d y_1, d x) eq f_1 (x, y_1, y_2, dots, y_n), + dots, + frac(d y_n, d x) eq f_n (x, y_1, y_2, dots, y_n)) +$ + +Решением системы называется совокупность $n$ функций + +$ +y_i eq psi_i (x), space i eq 1, 2, dots, n +$ + +таких, что при подстановке их в уравнения системы эти уравнения обращаются в тождества относительно $x$. При этом функции $psi_i (x)$ предполагаются непрерывно дифференцируемыми. + +#align(center)[ИЛИ] + +Нормальной системой дифференциальных уравнений порядка $n$ называется система уравнений вида + +$ +cases(dot(x)_1 eq f_1 (t, x_1, dots, x_n), + dots, + dot(x)_n eq f_n (t, x_1, dots, x_n)). +$ + +Если ввести в рассмотрение векторы + +$ +r eq vec(x_1, dots, x_n), space.quad f(t, r) eq vec(f_1 (t, r), dots, f_n (t, r)), +$ + +то систему можно компактно записать в виде одного $n$-мерного уравнения + +$ +dot(r) eq f(t, r). +$ + +Вектор-функция $phi$ - решение системы на $(a, b)$, если + +$ +phi in C^1 ((a, b) arrow RR_n); \ +dot(phi) (t) eq.triple f(t, phi(t)) " на " (a, b). +$ + +#line(length: 100%) + +18. Линейная однородная и неоднородная система ДУ + +Линейной системой дифференциальных уравнений называют систему вида + +$ +dot(r) eq P(t) r plus q(t), +$ + +где $P in M_n (C(a, b)), space q in C((a, b) arrow RR^n)$. + +Если $q eq.triple 0$ на $(a, b)$, то система, то есть + +$ +dot(r) eq P(t) r, +$ + +называется однородной, в противном случае -- неоднородной. + +#line(length: 100%) + +19. Функция оригинал + +Функцией-оригиналом называется комплекснозначная функция $f(t)$ вещественной переменной $t$, удовлетворяющая следующим условиям: + +- $f(t) eq 0$, если $t lt 0$; +- $f(t)$ интегрируема на любом конечном интервале оси $t$; +- с возрастанием $t$ модуль функции $f(t)$ растет не быстрее некоторой показательной функции, то есть существуют числа $M gt 0$ и $s_0 gt.eq 0$ такие, что для всех $t$ имеем: + +$ +|f(t)| lt.eq M e^(s_0 t). +$ + +#line(length: 100%) + +20. Преобразование Лапласа + +Преобразованием Лапласа $L$ функции-оригинала $f(t)$, заданной на $[0, infinity)$, называется преобразование вида: + +$ +(L f) (p) eq F(p) eq integral_0^infinity f(t) e^(- p t) d t, +$ + +где образ функции $f$ будем обозначать за $F(p)$. Функцию $F(p)$ называют изображением функции-оригинала $f(t)$. + +#pagebreak() +#align(center)[=== Теоремы] + +1. О существовании решения ДУ +2. Решение однородного дифференциального уравнения +3. О решении линейного однородного уравнения +4. Метод Лагранжа (вариации произвольной постоянной) +5. Метод Бернулли +6. О полном дифференциале +7. Об интегрирующем множителе +8. О существовании решения ДУ высших порядков +9. Замены для уравнений, допускающих понижение порядка +10. Свойства решений линейного однородного ДУ +11. Необходимое условие линейной зависимости решений +12. Достаточное условие линейной зависимости решений +13. О базисе пространства решений +14. Общее решение линейного неоднородного ДУ +15. Принцип суперпозиции +16. Метод вариации произвольных постоянных +17. О ФСР для различных вещественных корней характеристического многочлена +18. О ФСР для кратных вещественных корней характеристического многочлена +19. Линейное однородное ДУ второго порядка с постоянными коэффициентами +20. Метод неопределенных коэффициентов +21. Метод исключения для решения системы ДУ +22. Метод Эйлера для решения однородных линейных систем ДУ при простых собственных числах +23. Метод Эйлера для решения однородных линейных систем ДУ при кратных собственных числах +24. Общее решение линейной неоднородной системы ДУ +25. Свойства преобразования Лапласа +26. О дифференцировании изображения +27. О дифференцировании оригинала +28. Об интегрировании оригинала +29. Преобразования Лапласа простейших функций diff --git a/exam/theory.pdf b/exam/theory.pdf new file mode 100644 index 0000000..c6dfb38 Binary files /dev/null and b/exam/theory.pdf differ