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Консультация № 186254
30.05.2012, 16:51
202.44 руб.
0
2
2
Здравствуйте! У меня возникли сложности с таким вопросом:
Заранее благодарен!!!
Обсуждение
31.05.2012, 00:33
общий
это ответ
Здравствуйте, sereggg!
1.
Для заданного диф.уравнения характеристическое уравнение имеет вид
Исследуем 3 случая:
1)
Корни характеристического уравнения:
Общее решение диф.уравнения:
Используем граничные условия:
Полученная однородная система линеных уравнений имеет нетривиальное решение, если ее определитель равен нулю:
С учетом найденного значения система перепишется в виде:
Из первого уравнения определяем, что С2=0, тогда из второго уравнения следует, что С1 - любое.
Итак, собственным числам задачи Штурма-Лиувилля
соответствует бесконечное множество собственных функций
2)
Диф.уравнение примет вид
его решением будет 
Используя граничные условия
находим только тривиальное решение 
3)
Корни характеристического уравнения:
Общее решение диф.уравнения:
Используем граничные условия:
Определитель этой однородной системы
система имеет только тривиальное решение. То есть опять получим
Таким образом, искомые нетривиальные решения задачи Штурма-Лиувилля имеют вид
1.
Для заданного диф.уравнения характеристическое уравнение имеет вид
Исследуем 3 случая:
1)
Корни характеристического уравнения:
Общее решение диф.уравнения:
Используем граничные условия:
Полученная однородная система линеных уравнений имеет нетривиальное решение, если ее определитель равен нулю:
С учетом найденного значения система перепишется в виде:
Из первого уравнения определяем, что С2=0, тогда из второго уравнения следует, что С1 - любое.
Итак, собственным числам задачи Штурма-Лиувилля
соответствует бесконечное множество собственных функций2)
Диф.уравнение примет вид
его решением будет Используя граничные условия
находим только тривиальное решение 3)
Корни характеристического уравнения:
Общее решение диф.уравнения:
Используем граничные условия:
Определитель этой однородной системы
система имеет только тривиальное решение. То есть опять получим
Таким образом, искомые нетривиальные решения задачи Штурма-Лиувилля имеют вид
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