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Дифференциальное уравнение + +Обыкновенным дифференциальным уравнением первого порÑдка называют уравнение вида + +$ +F(x, y, y') eq 0. +$ + +#align(center)[ИЛИ] + +Дифференциальным уравнением называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ðµ, ÑвÑзывающее незавиÑимую переменную $x$, иÑкомую функцию $y eq y(x)$ и её производные $y′, y′′, dots , y(n)$. + +#line(length: 100%) + +2. Решение дифференциального уравнениÑ, общее решение + +Ð¤ÑƒÐ½ÐºÑ†Ð¸Ñ $phi$ - решение уравнениÑ, еÑли + +$ +phi in C^1 (a, b); \ +F(x, phi(x), phi'(x)) eq.triple 0 " на " (a, b) +$ + +Другими Ñловами, решением ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð°Ð·Ñ‹Ð²Ð°ÑŽÑ‚ гладкую функцию $phi$, определённую на интервале $(a, b)$, подÑтановка которой вмеÑто $y$ обращает уравнение в тождеÑтво на $(a, b)$. + +Общим решением ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð°Ð·Ñ‹Ð²Ð°ÑŽÑ‚ множеÑтво вÑех его решений. + +#align(center)[ИЛИ] + +Ð¤ÑƒÐ½ÐºÑ†Ð¸Ñ $y eq phi(x)$ ÑвлÑетÑÑ Ñ€ÐµÑˆÐµÐ½Ð¸ÐµÐ¼ дифференциального уравнениÑ, еÑли её подÑтановка в уравнение обращает его в тождеÑтво. + +#line(length: 100%) + +3. Задача Коши + +Задачей Коши или начальной задачей Ð´Ð»Ñ Ð½Ð¾Ñ€Ð¼Ð°Ð»ÑŒÐ½Ð¾Ð³Ð¾ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ + +$ +y' eq f(x, y) +$ + +называют задачу Ð½Ð°Ñ…Ð¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÐµÐ³Ð¾ решениÑ, удовлетворÑющего начальному уÑловию + +$ +y(x_0) eq y_0. +$ + +Пара чиÑел $(x_0, y_0)$ при Ñтом называетÑÑ Ð½Ð°Ñ‡Ð°Ð»ÑŒÐ½Ñ‹Ð¼Ð¸ данными. + +#line(length: 100%) + +4. Уравнение Ñ Ñ€Ð°Ð·Ð´ÐµÐ»ÑющимиÑÑ Ð¿ÐµÑ€ÐµÐ¼ÐµÐ½Ð½Ñ‹Ð¼Ð¸ + +Уравнение в дифференциалах вида + +$ +P(x) d x plus Q(y) d y eq 0 +$ + +называют уравнением Ñ Ñ€Ð°Ð·Ð´ÐµÐ»Ñ‘Ð½Ð½Ñ‹Ð¼Ð¸ переменными. + +Такое название мотивировано тем, что каждое его Ñлагаемое завиÑит 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 Ñ Ñ€Ð°Ð·Ð´ÐµÐ»ÑющимиÑÑ Ð¿ÐµÑ€ÐµÐ¼ÐµÐ½Ð½Ñ‹Ð¼Ð¸. + +#align(center)[ИЛИ] + +ЕÑли уравнение $Phi(x, y, y') eq 0$ Ñ Ð¿Ð¾Ð¼Ð¾Ñ‰ÑŒÑŽ алгебраичеÑких преобразований удаетÑÑ Ð¿Ñ€Ð¸Ð²ÐµÑти 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 называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸ÐµÐ¼ Ñ Ñ€Ð°Ð·Ð´ÐµÐ»ÑющимиÑÑ Ð¿ÐµÑ€ÐµÐ¼ÐµÐ½Ð½Ñ‹Ð¼Ð¸. + +#line(length: 100%) + +5. ÐžÐ´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ð°Ñ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ñ + +Ð¤ÑƒÐ½ÐºÑ†Ð¸Ñ $F(x, y)$ называетÑÑ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ð¾Ð¹ функцией Ñтепени $alpha$, еÑли при вÑех допуÑтимых $t$, $x$ и $y$ верно равенÑтво + +$ +F(t x, t y) eq t^alpha F(x, y). +$ + +Пример однородных функций: $x plus y plus z$ (первой Ñтепени), $x^2 plus 3 x y plus y^2$ (второй Ñтепени), $y/x cos x/y$ (нулевой Ñтепени), $frac(sqrt(x plus y), x^2 plus y^2)$ (Ñтепени $-3/2$). + +#line(length: 100%) + +6. Однородное ДУ первого порÑдка + +ПуÑть $P$ и $Q$ - однородные функции одинаковой Ñтепени. Тогда уравнение вида + +$ +P(x, y) d x plus Q(x, y) d y eq 0 +$ + +называетÑÑ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ñ‹Ð¼ уравнением. + +#align(center)[ИЛИ] + +Дифференциальное уравнение первого порÑдка называетÑÑ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ñ‹Ð¼, еÑли его можно привеÑти к виду: + +$ +y' eq f(y/x) +$ + +#line(length: 100%) + +7. Линейное ДУ первого порÑдка + +Дифференциальное уравнение вида + +$ +y' eq p(x) y plus q(x), +$ + +называетÑÑ Ð»Ð¸Ð½ÐµÐ¹Ð½Ñ‹Ð¼ уравнением первого порÑдка. + +Ðазвание линейное мотивировано тем, что оно ÑоÑтавлено из многочленов первой Ñтепени по отношению к Ñимволам $y$ и $y'$. + +#align(center)[ИЛИ] + +Линейным уравнением первого порÑдка называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ðµ 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). +$ + +ОтÑюда видно, что замена $z eq y^(1 minus alpha)$ Ñводит уравнение к линейному. + +#align(center)[ИЛИ] + +Уравнением Бернулли называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ðµ 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, еÑли ÑущеÑтвует Ñ‚Ð°ÐºÐ°Ñ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ñ $u$, что + +$ +d u eq P(x, y) d x plus Q(x, y) d y, +$ + +то еÑть $u'_x eq P, u'_y eq Q$. + +#align(center)[ИЛИ] + +Дифференциальное уравнение вида + +$ +M(x, y) d x plus N(x, y) d x eq 0 +$ + +называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸ÐµÐ¼ в полных дифференциалах, еÑли его Ð»ÐµÐ²Ð°Ñ Ñ‡Ð°Ñть предÑтавлÑет Ñобой 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. +$ + +УÑловие того, что $M d x plus N d y$ предÑтавлÑет Ñобой полный дифференциал: + +$ +frac(partial M, partial y) eq frac(partial N, partial x). +$ + +#line(length: 100%) + +10. ОÑобое решение ДУ + +Решение $y eq phi(x)$ дифференциального ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ + +$ +Phi(x, y, y') eq 0 +$ + +называетÑÑ Ð¾Ñобым, еÑли в каждой его точке нарушаетÑÑ ÑвойÑтво единÑтвенноÑти, то еÑть еÑли через каждую его точку ($x_0, y_0$) кроме Ñтого Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð¿Ñ€Ð¾Ñ…Ð¾Ð´Ð¸Ñ‚ и другое решение, имеющее в точке ($x_0, y_0$) ту же каÑательную, что и решение $y eq phi(x)$, но не Ñовпадающее Ñ Ð½Ð¸Ð¼ в Ñколь угодно малой окреÑтноÑти ($x_0, y_0$). График оÑобого Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð±ÑƒÐ´ÐµÐ¼ называть оÑобой интегральной кривой уравнениÑ. + +#align(center)[ИЛИ] + +Решение $phi$ на $(a, b)$ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ $y' eq f(x, y)$ называетÑÑ Ð¾Ñобым, еÑли Ð´Ð»Ñ Ð»ÑŽÐ±Ð¾Ð¹ точки $x_0 in (a, b)$ найдетÑÑ Ñ€ÐµÑˆÐµÐ½Ð¸Ðµ $psi$ того же уравнениÑ, такое что + +$ +phi(x_0) eq psi(x_0) +$ + +при Ñтом $phi eq.triple psi$ в любой Ñколь угодно малой окреÑтноÑти точки $x_0$. + +Более кратко Ñто выражают Ñловами: Ð¸Ð½Ñ‚ÐµÐ³Ñ€Ð°Ð»ÑŒÐ½Ð°Ñ ÐºÑ€Ð¸Ð²Ð°Ñ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ $y' eq f(x, y)$ ÑвлÑетÑÑ Ð¾Ñобой, еÑли в каждой её точке нарушаетÑÑ ÐµÐ´Ð¸Ð½ÑтвенноÑть Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°Ñ‡Ð¸ Коши. + +#line(length: 100%) + +11. ДУ выÑшего порÑдка, задача Коши Ð´Ð»Ñ Ð½ÐµÐ³Ð¾ + +Дифференциальным уравнением $n$-го порÑдка называют уравнение вида + +$ +F(x, y, y', dots, y^((n))) eq 0. +$ + +Ð¤ÑƒÐ½ÐºÑ†Ð¸Ñ $phi$ -- решение ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð° $(a, b)$, еÑли + +$ +phi in C^n (a, b); \ +F(x, phi(x), phi'(x), dots, phi^((n))(x)) eq.triple 0 " на " (a, b). +$ + +КаноничеÑким уравнением будем называть уравнение + +$ +y^((n)) eq f(x, y, y', dots, y^((n minus 1))), +$ + +разрешённое отноÑительно Ñтаршей производной. + +Задачей Коши Ð´Ð»Ñ ÐºÐ°Ð½Ð¾Ð½Ð¸Ñ‡ÐµÑкого ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð°Ð·Ñ‹Ð²Ð°ÑŽÑ‚ задачу Ð½Ð°Ñ…Ð¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÐµÐ³Ð¾ решениÑ, удовлетворÑющего начальным уÑловиÑм + +$ +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)). +$ + +Ðабор чиÑел $(x_0, y_0, y'_0, dots, y_0^((n minus 1)) )$ при Ñтом называют начальными данными. + +#align(center)[ИЛИ] + +Обыкновенное дифференциальное уравнение $n$-го порÑдка имеет вид + +$ +Phi(x, y, y', y'', dots, y^((n))) eq 0, +$ + +или в решенном отноÑительно Ñтаршей производной $y^((n))$, вид + +$ +y^((n)) eq f(x, y, y', y'', dots, y^((n minus 1))). +$ + +Ð’ÑÑÐºÐ°Ñ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ñ $y(x)$, Ð¸Ð¼ÐµÑŽÑ‰Ð°Ñ Ð½ÐµÐ¿Ñ€ÐµÑ€Ñ‹Ð²Ð½Ñ‹Ðµ производные вплоть до $n$-го порÑдка и удовлетворÑÑŽÑ‰Ð°Ñ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸ÑŽ, называетÑÑ Ñ€ÐµÑˆÐµÐ½Ð¸ÐµÐ¼ Ñтого уравнениÑ, а Ñама задача Ð½Ð°Ñ…Ð¾Ð¶Ð´ÐµÐ½Ð¸Ñ Ñ€ÐµÑˆÐµÐ½Ð¸Ð¹ дифференциального ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð°Ð·Ñ‹Ð²Ð°ÐµÑ‚ÑÑ Ð·Ð°Ð´Ð°Ñ‡ÐµÐ¹ Ð¸Ð½Ñ‚ÐµÐ³Ñ€Ð¸Ñ€Ð¾Ð²Ð°Ð½Ð¸Ñ Ð´Ð¸Ñ„Ñ„ÐµÑ€ÐµÐ½Ñ†Ð¸Ð°Ð»ÑŒÐ½Ð¾Ð³Ð¾ уравнениÑ. + +#line(length: 100%) + +12. Линейное ДУ $n$-го порÑдка. Однородное, неоднородное + +Линейным дифференциальным уравнением порÑдка $n$ называетÑÑ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ðµ 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 $p_0, p_1, dots, p_(n minus 1), q in C(a, b)$. + +ЕÑли $q eq.triple 0$ на $(a, b)$, то уравнение, то еÑть + +$ +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, +$ + +называетÑÑ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ñ‹Ð¼, в противном Ñлучае -- 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 +$ + +называетÑÑ Ð»Ð¸Ð½ÐµÐ¹Ð½Ñ‹Ð¼ однородным дифференциальным уравнением $n$-го порÑдка. + +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) +$ + +называетÑÑ Ð»Ð¸Ð½ÐµÐ¹Ð½Ñ‹Ð¼ неоднородным дифференциальным уравнением $n$-го порÑдка. + +#line(length: 100%) + +13. Ð›Ð¸Ð½ÐµÐ¹Ð½Ð°Ñ Ð½ÐµÐ·Ð°Ð²Ð¸ÑимоÑть функций + +#line(length: 100%) + +14. Определитель ВронÑкого + +Определителем ВронÑкого (или вронÑкианом) 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. Ð¤ÑƒÐ½Ð´Ð°Ð¼ÐµÐ½Ñ‚Ð°Ð»ÑŒÐ½Ð°Ñ ÑиÑтема решений + +Фундаментальной ÑиÑтемой решений ÑиÑтемы уравнений называетÑÑ ÑовокупноÑть её $n$ линейно незавиÑимых решений. + +#align(center)[ИЛИ] + +Любой набор из $n$ линейно незавиÑимых решений $y_1 (x), y_2 (x), dots, y_n (x)$ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ $y^((n)) plus a_1 (x) y^((n minus 1)) plus dots plus a_(n minus 1) (x) y' plus a_n (x) y eq 0$ называетÑÑ Ñ„ÑƒÐ½Ð´Ð°Ð¼ÐµÐ½Ñ‚Ð°Ð»ÑŒÐ½Ð¾Ð¹ ÑиÑтемой решений Ñтого уравнениÑ. + +#line(length: 100%) + +16. ХарактериÑтичеÑкий 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 +$ + +называетÑÑ Ñ…Ð°Ñ€Ð°ÐºÑ‚ÐµÑ€Ð¸ÑтичеÑким многочленом ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ $y^((n)) plus a_(n minus 1) y^((n minus 1)) plus dots plus a_1 dot(y) plus a_0 y eq f(t)$, а его корни -- характериÑтичеÑкими чиÑлами того же уравнениÑ. + +#line(length: 100%) + +17. СиÑтема ДУ, решение ÑиÑтемы + +СиÑтема дифференциальных уравнений -- Ñто набор дифференциальных уравнений, решаемых ÑовмеÑтно. Решение ÑиÑтемы -- Ñто набор функций, который удовлетворÑет вÑем уравнениÑм ÑиÑтемы. Ð¢Ð°ÐºÐ°Ñ Ñ„Ð¾Ñ€Ð¼Ð° запиÑи ÑиÑтемы называетÑÑ Ð½Ð¾Ñ€Ð¼Ð°Ð»ÑŒÐ½Ð¾Ð¹ 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 ÑиÑтемы называетÑÑ ÑовокупноÑть $n$ функций + +$ +y_i eq psi_i (x), space i eq 1, 2, dots, n +$ + +таких, что при подÑтановке их в ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ ÑиÑтемы Ñти ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ Ð¾Ð±Ñ€Ð°Ñ‰Ð°ÑŽÑ‚ÑÑ Ð² тождеÑтва отноÑительно $x$. При Ñтом функции $psi_i (x)$ предполагаютÑÑ Ð½ÐµÐ¿Ñ€ÐµÑ€Ñ‹Ð²Ð½Ð¾ дифференцируемыми. + +#align(center)[ИЛИ] + +Ðормальной ÑиÑтемой дифференциальных уравнений порÑдка $n$ называетÑÑ ÑиÑтема 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)). +$ + +ЕÑли ввеÑти в раÑÑмотрение 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 ÑиÑтему можно компактно запиÑать в виде одного $n$-мерного ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ + +$ +dot(r) eq f(t, r). +$ + +Вектор-Ñ„ÑƒÐ½ÐºÑ†Ð¸Ñ $phi$ - решение ÑиÑтемы на $(a, b)$, еÑли + +$ +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. Ð›Ð¸Ð½ÐµÐ¹Ð½Ð°Ñ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ð°Ñ Ð¸ Ð½ÐµÐ¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ð°Ñ ÑиÑтема ДУ + +Линейной ÑиÑтемой дифференциальных уравнений называют ÑиÑтему вида + +$ +dot(r) eq P(t) r plus q(t), +$ + +где $P in M_n (C(a, b)), space q in C((a, b) arrow RR^n)$. + +ЕÑли $q eq.triple 0$ на $(a, b)$, то ÑиÑтема, то еÑть + +$ +dot(r) eq P(t) r, +$ + +называетÑÑ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ð¾Ð¹, в противном Ñлучае -- неоднородной. + +#line(length: 100%) + +19. Ð¤ÑƒÐ½ÐºÑ†Ð¸Ñ Ð¾Ñ€Ð¸Ð³Ð¸Ð½Ð°Ð» + +Функцией-оригиналом называетÑÑ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½Ð¾Ð·Ð½Ð°Ñ‡Ð½Ð°Ñ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ñ $f(t)$ вещеÑтвенной переменной $t$, удовлетворÑÑŽÑ‰Ð°Ñ Ñледующим уÑловиÑм: + +- $f(t) eq 0$, еÑли $t lt 0$; +- $f(t)$ интегрируема на любом конечном интервале оÑи $t$; +- Ñ Ð²Ð¾Ð·Ñ€Ð°Ñтанием $t$ модуль функции $f(t)$ раÑтет не быÑтрее некоторой показательной функции, то еÑть ÑущеÑтвуют чиÑла $M gt 0$ и $s_0 gt.eq 0$ такие, что Ð´Ð»Ñ Ð²Ñех $t$ имеем: + +$ +|f(t)| lt.eq M e^(s_0 t). +$ + +#line(length: 100%) + +20. Преобразование ЛаплаÑа + +Преобразованием ЛаплаÑа $L$ функции-оригинала $f(t)$, заданной на $[0, infinity)$, называетÑÑ Ð¿Ñ€ÐµÐ¾Ð±Ñ€Ð°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ 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 ÑущеÑтвовании Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð”Ð£ +2. Решение однородного дифференциального ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ +3. О решении линейного однородного ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ñ +4. Метод Лагранжа (вариации произвольной поÑтоÑнной) +5. Метод Бернулли +6. О полном дифференциале +7. Об интегрирующем множителе +8. О ÑущеÑтвовании Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð”Ð£ выÑших порÑдков +9. Замены Ð´Ð»Ñ ÑƒÑ€Ð°Ð²Ð½ÐµÐ½Ð¸Ð¹, допуÑкающих понижение порÑдка +10. СвойÑтва решений линейного однородного ДУ +11. Ðеобходимое уÑловие линейной завиÑимоÑти решений +12. ДоÑтаточное уÑловие линейной завиÑимоÑти решений +13. О базиÑе проÑтранÑтва решений +14. Общее решение линейного неоднородного ДУ +15. Принцип Ñуперпозиции +16. Метод вариации произвольных поÑтоÑнных +17. О ФСР Ð´Ð»Ñ Ñ€Ð°Ð·Ð»Ð¸Ñ‡Ð½Ñ‹Ñ… вещеÑтвенных корней характериÑтичеÑкого многочлена +18. О ФСР Ð´Ð»Ñ ÐºÑ€Ð°Ñ‚Ð½Ñ‹Ñ… вещеÑтвенных корней характериÑтичеÑкого многочлена +19. Линейное однородное ДУ второго порÑдка Ñ Ð¿Ð¾ÑтоÑнными коÑффициентами +20. Метод неопределенных коÑффициентов +21. Метод иÑÐºÐ»ÑŽÑ‡ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ñ€ÐµÑˆÐµÐ½Ð¸Ñ ÑиÑтемы ДУ +22. Метод Эйлера Ð´Ð»Ñ Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ñ‹Ñ… линейных ÑиÑтем ДУ при проÑтых ÑобÑтвенных чиÑлах +23. Метод Эйлера Ð´Ð»Ñ Ñ€ÐµÑˆÐµÐ½Ð¸Ñ Ð¾Ð´Ð½Ð¾Ñ€Ð¾Ð´Ð½Ñ‹Ñ… линейных ÑиÑтем ДУ при кратных ÑобÑтвенных чиÑлах +24. Общее решение линейной неоднородной ÑиÑтемы ДУ +25. СвойÑтва Ð¿Ñ€ÐµÐ¾Ð±Ñ€Ð°Ð·Ð¾Ð²Ð°Ð½Ð¸Ñ Ð›Ð°Ð¿Ð»Ð°Ñа +26. О дифференцировании Ð¸Ð·Ð¾Ð±Ñ€Ð°Ð¶ÐµÐ½Ð¸Ñ +27. О дифференцировании оригинала +28. Об интегрировании оригинала +29. 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