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Консультация № 186149
22.05.2012, 16:59
300.34 руб.
23.05.2012, 12:36
0
3
2
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Обсуждение
22.05.2012, 23:11
общий
проверьте пожалуйста условие 4-ой задачи.
с одной стороны 0<=r<2, c другой стороны u(r=1).
с одной стороны 0<=r<2, c другой стороны u(r=1).
23.05.2012, 10:12
общий
это ответ
Здравствуйте, sereggg!
2.
Имеем уравнение в частных производных, которое является линейным однородным уравнением с постоянными коэффициентами.
а11=1, а12=1, а22=1; (а12)2-а11[$183$]а22=0 [$8658$]
по классификации это уравнение параболического типа.
Характеристическое уравнение будет иметь вид
, или 
.
Общий интеграл этого уравнения: y-x=C.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Преобразуем производные к новым переменным, используя следующие формулы:
Получим
Подставив их в уравнение, имеем:
После упрощения, приходим к канонической форме уравнения:
Решим полученное уравнение:
Возвращаясь к старым переменным, получим искомое общее решение:
где f и g - произвольные функции указанных аргументов.
3.
а11=3, а12=8, а22=16; (а12)2-а11[$183$]а22=64-48=16>0 [$8658$]
по классификации это уравнение гиперболического типа.
Характеристическое уравнение будет иметь вид
.
Получим два уравнения
Общие интегралы этих уравнений: y-4x=C1, 3y-4x=C2.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Пересчитаем производные в новых переменных:
Подставив их в уравнение, имеем:
После упрощения, приходим к канонической форме уравнения:
Интегрируя полученное уравнение, находим:
Возвращаясь к старым переменным, получим искомое общее решение:
где f и g - произвольные функции указанных аргументов.
2.
Имеем уравнение в частных производных, которое является линейным однородным уравнением с постоянными коэффициентами.
а11=1, а12=1, а22=1; (а12)2-а11[$183$]а22=0 [$8658$]
по классификации это уравнение параболического типа.
Характеристическое уравнение будет иметь вид
, или .Общий интеграл этого уравнения: y-x=C.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Преобразуем производные к новым переменным, используя следующие формулы:
Получим
Подставив их в уравнение, имеем:
После упрощения, приходим к канонической форме уравнения:
Решим полученное уравнение:
Возвращаясь к старым переменным, получим искомое общее решение:
где f и g - произвольные функции указанных аргументов.
3.
а11=3, а12=8, а22=16; (а12)2-а11[$183$]а22=64-48=16>0 [$8658$]
по классификации это уравнение гиперболического типа.
Характеристическое уравнение будет иметь вид
.Получим два уравнения
Общие интегралы этих уравнений: y-4x=C1, 3y-4x=C2.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Пересчитаем производные в новых переменных:
Подставив их в уравнение, имеем:
После упрощения, приходим к канонической форме уравнения:
Интегрируя полученное уравнение, находим:
Возвращаясь к старым переменным, получим искомое общее решение:
где f и g - произвольные функции указанных аргументов.
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