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ТЕОРЕТИЧЕСКИЕ ВОПРОСЫ Основные понятия теории дифференциальных уравнений

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§ 5.1. ТЕОРЕТИЧЕСКИЕ ВОПРОСЫ

  1.  Основные понятия теории дифференциальных уравнений. Задача Коши для дифференциального уравнения первого порядка. Формулировка теоремы существования и единственности решения задачи Коши.
  2.  Дифференциальные уравнения первого порядка: с разделяющимися переменными, однородные и приводящиеся к ним.
  3.  Линейные уравнения первого порядка, уравнение Бернулли.
  4.  Уравнения в полных дифференциалах.
  5.  Приближенное интегрирование дифференциальных уравнений первого порядка методом изоклин.
  6.  Дифференциальные уравнения высших порядков. Задача Коши. Формулировка теоремы существования и единственности решения задачи Коши. Общее и частное решения. Общий и частный интегралы.
  7.  Дифференциальные уравнения, допускающие понижение порядка.
  8.  Линейный дифференциальный оператор, его свойства. Линейное однородное дифференциальное уравнение, свойства его решений.
  9.  Линейно-зависимые и линейно-независимые системы функций. Необходимое условие линейной зависимости системы функций.

  1.  Условие линейной независимости решений линейного однородного дифференциального уравнения.
  2.  Линейное однородное дифференциальное уравнение. Фундаментальная система решений. Структура общего решения.
  3.  Линейное неоднородное дифференциальное уравнение. Структура общего решения.
  4.  Метод Лагранжа вариации произвольных постоянных.
  5.  Линейные однородные дифференциальные уравнения с постоянными коэффициентами (случай простых корней характеристического уравнения).
  6.  Линейные однородные дифференциальные уравнении с постоянными коэффициентами (случай кратных корней характеристического уравнения).
  7.  Линейные неоднородные дифференциальные уравнения с постоянными коэффициентами. Метод подбора.

§ 5.2. ТЕОРЕТИЧЕСКИЕ УПРАЖНЕНИЯ

  1.  Пусть  —решение дифференциального уравнения . Показать, что введение новой искомой функции  приводит к дифференциальному уравнению, допускающему понижение порядка.
  2.  Написать уравнение линии, на которой могут находиться точки перегиба графиков решений уравнения .
  3.  Написать уравнение линии, на которой могут находиться точки графиков решений уравнения , соответствующие максимумам и минимумам.

Как отличить максимум от минимума?

  1.  Линейное дифференциальное уравнение останется линейным при замене независимой переменной , где функция  произвольная, но дифференцируемая достаточное число раз. Доказать это утверждение для линейного дифференциального уравнения второго порядка.
  2.  Доказать, что линейное дифференциальное уравнение остается линейным при преобразовании искомой функции

Здесь —новая искомая функция , и  —произвольные, но достаточное число раз дифференцируемые функции.

6) Составить общее решение уравнения , если известно ненулевое частное решение  этого уравнения.

7) Показать, что произвольные дважды дифференцируемые функции  и  являются решениями линейного дифференциального уравнения

8) Составить однородное линейное дифференциальное уравнение второго порядка, имеющее .

Показать, что функции и  линейно независимы в интервале .

Убедиться в том, что определитель Вронского для этих функций равен нулю в точке . Почему это не противоречит необходимому условию линейной независимости системы решений линейного однородного дифференциального уравнения?

9) Найти общее решение неоднородного линейного дифференциального уравнения второго порядка, если известны три линейно-независимые частные его решения , , ?,.

  1.  Доказать, что для того чтобы любое решение линейного однородного дифференциального уравнения с постоянными коэффициентами удовлетворяло условию, , необходимо и достаточно, чтобы все корни характеристического уравнения имели отрицательные действительные части.

§ 5.3. РАСЧЕТНЫЕ ЗАДАНИЯ

Задача 1. Найти общий интеграл дифференциального уравнения. (Ответ представить в виде

1.                                              2.                           3.                            4.                               5.                            6.                         7.                               8.                                          9.                           10.                          11.                              

.                              13.                     14.                         15.                              16.                                             17.                 18.                                             19.                                           20.                             21.                           22.                                    23.                                            24.                         25.                          26.                           27.                                            28.                         29.                              30.                            31.                         

Задача 2. Найти общий интеграл дифференциального уравнения.

1.                                              2.                                              3.                                                           4.                                              5.                                               6.                                               7.                                                         8.                                               9.                                            10.                                          11.                                         12.                                          13.                                          14.                                              15.                                               

.                                          17.                                        18.                                        19.                                        20.                                        21.                                          22.                                        23.                                          24.                                           25.                                          26.                                          27.                                          28.                                         29.                                          30.                                        31.                         

Задача 3. Найти общий интеграл дифференциального уравнения.

1.                               2.                               3.                                 4.                               5.                                    6.                                 7.                               8.                                 9.                               10.

.                                                               12.                                    13.                                 14.                                15.                                  16.                               17.                                18.                                    19.                               20.                    

21.                                 22.                               23.                                 24.                               25.                                   26.                                 27.                           28.                                29.                                 30.                               31. .

Задача 4. Найти решение задачи Коши.

1.                                              2.                                              3.                                               4.                                              5.                                               6.                                               

.                                               8.                                               9.                                              10.                                           11.                                           12.                                           13.                                           14.                                              15.                                               16.                                           17.                                           18.                                           19.                                           20.                                            21.                                            22.                                           23.                                           24.                                                25.                                                   26.                                                  27.                                                28.                                                29.                                                30.                                                  31.        

Задача 5. Решить задачу Коши.

.                                                                                                              2.                                                                                                                               3.                                                                                                                   4.                                                                                                                 5.                                                                                                              6.                                                                                                              7.                                                                                                                8.                                                                                                                      9.                                                                                                                10.                                                                                                              11.                                                                                                              12.                                                                                                                13.                                                                                                                      14.                                                                                                              15.                                                                                                              16.                                                                                                              17.                                                                                                              18.                                                                                                                   19.                                                                                                              20.                                                                                                              21.                                                                                                              22.                                                                                                              23.                                                                                                                    24.                                                                                                            25.                                                                                                              26.                                                                                                              27.                                                                                                              28.                                                                                                              29.                                                                                                              30.                                                                                                              31.                 

Задача 6. Найти решение задачи Коши.

.                                                                                                                 2.                                                                                                                       3.                                                                                                                           4.                                                                                                                 5.                                                                                                                 6.                                                                                                                 7.                                                                                                                      8.                                                                                                                 9.                                                                                                                10.                                                                                                              11.                                                                                                              12.                                                                                                                13.                                                                                                              14.                                                                                                                       15.                                                                                                                           16.                                                                                                              17.                                                                                                                   18.                                                                                                                       19.                                                                                                              20.                                                                                                              21.                                                                                                              22.                                                                                                                        23.                                                                                                               24.                                                                                                              25.                                                                                                                             26.                                                                                                                  27.                                                                                                                              28.                                                                                                              29.                                                                                                              30.                                                                                                              31.                          

     Задача 7. Найти общий интеграл дифференциального уравнения.

1.                                                                                                                        2.                                                                                                                        3.                                                                                                                         4.                                                                                                                               5.                                                                                                                          6.                                                                                                                         7.                                                                                                                        8.                                                                                                                          9.                                                                                                                      10.                                                                                                                     11.                                                                                                                     12.                                                                                                                      13.                                                                                                                         14.                                                                                                                                 15.                                                                                                                               16.                                                                                                                        17.                                                                                                                    18.                                                                                                                     19.                                                                                                                      20.                                                                                                                      21.                                                                                                                    22.                                                                                                                    23.                                                                                                                    24.                                                                                                                    25.                                                                                                                     26.                                                                                                                    27.                                                                                                                       28.                                                                                                                    29.                                                                                                                    30.                                                                                                                      31.                             

 Задача 8. Для данного дифференциального уравнения методом изоклин построить интегральную кривую, проходящую через точку

1.                                              2.                                              3.                                               4.                                                5.                                               6.                                               7.                                               8.                                               9.                                            10.                                          11.                                         12.                                          13.                                          14.                                              15.                                               

.                                     17.                                        18.                                          19.                                        20.                                        21.                                              22.                                                23.                                               24.                                                25.                                                   26.                                           27.                                                28.                                                29.                                                30.                                                  31.        

Задача 9. Найти линию, проходящую через точку  и обладающую тем свойством, что в любой ее точке  нормальный вектор  с концом на оси имеет длину, равную , и образует острый угол с положительным направлением оси .

1.                                                 2.                                                         3.

.                                               5.

Найти линию, проходящую через точку , если отрезок любой ее нормали, заключенный между осями координат, делится точкой линии в отношении   (считая от оси ).

6.                                                 7.                                                         8.

.                                               10.

Найти линию, проходящую через точку , если отрезок любой ее касательной между точкой касания и осью делится на точке пересечения с осью абсцисс  в отношении   (считая от оси ).

11.                                                 12.                                                         13.

14.                                               15.

Найти линию, проходящую через точку , если отрезок любой ее касательной, заключенный между осями координат, делится в точке касания в  отношении   (считая от оси ).

16.                                                 17.                                                         18.                                     19.                                               20.

Найти линию, проходящую через точку  и обладающую тем свойством, что в любой ее точке  касательный вектор  с концом на оси имеет проекцию на ось , обратно пропорциональную абсциссе точки . Коэффициент пропорциональности равен .

21.                                                                   24.                                                             22.                                                                  25.                                                                     23.

Найти линию, проходящую через точку  и обладающую тем свойством, что в любой ее точке  касательный вектор  с концом на оси имеет проекцию на ось , равную .

26.                                                 27.                                                             28.

29.                                                                     30.                                                    31.

Задача 10. Найти общее решение дифференциального уравнения.

1.                               2.                               3.                                 4.                               5.                                    6.                                 7.                               8.                                 9.                            10.                                       11.                 12.                                    

.                                 14.                                15.                                  16.                               17.                                18.                                    19.                               20.                                21.                                 22.                 23.                                 

.                               25.                                 26.                                 27.                      28.                               29.                                 30.                                 31.

       Задача 11. Найти решение задачи Коши.

1.                                                                                                                        2.                                                                                                                        3.                                                                                                                         4.                                                                                                                               5.                                                                                                                          6.                                                                                                                         7.                                                                                                                        8.                                                                                                                          9.                                                                                                                      10.                                                                                                                     11.                                                                                                                    12.                                                                                                                      13.                                                                                                                    14.                                                                                                                       15.                                                                                                                       16.                                                                                                                      17.                                                                                                                    18.                                                                                                                     19.                                                                                                                      20.                                                                                                                      21.                                                                                                                    22.                                                                                                                    23.                                                                                                                    24.                                                                                                                    25.                                                                                                                     26.                                                                                                                    27.                                                                                                                      28.                                                                                                                    29.                                                                                                                    30.                                                                                                                    31.     

Задача 12. Найти общее решение дифференциального уравнения.

1.                                   2.                                              3.                                               4.                                              5.                                               

.                                               7.                                               8.                                               9.                                                         10.                                          11.                                         12.                                          13.                                             14.                                              15.                                               16.                                          17.                                        18.                                        19.                                        20.                                        21.                                          22.                                        23.                                          24.                                                  25.                                          26.                                          27.                                          28.                                         29.                                          30.                                        31.                                                             

Задача 13. Найти общее решение дифференциального уравнения.

1.                                   2.                                              3.                                               4.                                              5.                                               6.                                               7.                                               8.                                               9.                                            10.                                          11.                                         12.                                          13.                                          14.                                              15.                                               16.  

.                                   18.                                              19.                                               20.                                              21.                                               22.                                               23.                                               24.                                               25.                                            26.                                          27.                                         28.                                          29.                                          30.                                              31.                                               

Задача 14. Найти общее решение дифференциального уравнения.

1.                                   2.                                              3.                                               4.                                              5.                                               6.                                               7.                                               8.                                               9.                                            10.                                          11.                                         12.                                          13.                                          14.                                              15.                                               16.                                          

17.                                        18.                                        19.                                        20.                                        21.                                       22.                                        23.                                          24.                                           25.                                          26.                                          27.                                          28.                                         29.                                          30.                                        31.                                                             

Задача 15. Найти общее решение дифференциального уравнения.

1.                                             2.                                              3.                                               4.                                              5.                                               6.                       7.                                             8.                                              9.                                               10.                                              11.                                               12.       

13.                                                               14.                                              15.                                               16.                                                          17.                                               18.                       19.                                                          20.                                              21.                                               22.                                                           23.                                               24.   

.                                               26.                                              27.                                               28.                                                      29.                                               30                                    31.                                                                                                                                                                                                                                                                                                                                                                                   

Задача 16. Найти решение задачи Коши.

.                                                                                                                 2.                                                                                                                 3.                                                                                                                 4.                                                                                                                 5.                                                                                                                 6.                                                                                                                 7.                                                                                                                  8.                                                                                                                 9.                                                                                                                10.                                                                                                              11.                                                                                                              12.                                                                                                                13.                                                                                                              14.                                                                                                              15.                                                                                                              16.                                                                                                              17.                                                                                                              18.                                                                                                              19.                                                                                                              20.                                                                                                              21.                                                                                                              22.                                                                                                              23.                                                                                                              24.                                                                                                              25.                                                                                                              26.                                                                                                              27.                                                                                                              28.                                                                                                              29.                                                                                                              30.                                                                                                              31.                          




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