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Консультация № 186241
29.05.2012, 17:22
300.01 руб.
0
2
2
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Заранее благодарен!!!
Обсуждение
29.05.2012, 18:57
общий
это ответ
Здравствуйте, sereggg!
Имеем уравнение в частных производных, которое является линейным однородным уравнением с постоянными коэффициентами.
а11=1, а12=-3, а22=9; (а12)2-а11[$183$]а22=9-9=0 [$8658$]
по классификации это уравнение параболического типа.
Характеристическое уравнение будет иметь вид
, или 
.
Общий интеграл этого уравнения: y+3x=C.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Преобразуем производные к новым переменным, используя следующие формулы:
Получим
Подставив их в исходное 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=16, а12=8, а22=3; (а12)2-а11·а22=64-48=16>0 ⇒
по классификации это уравнение гиперболического типа.
Характеристическое уравнение будет иметь вид
Получим два уравнения
Общие интегралы этих уравнений: 4y-3x=C1,4y-x=C2.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Пересчитаем производные 2-го порядка в новых переменных:
Подставив их в исходное 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.
Имеем уравнение в частных производных, которое является линейным однородным уравнением с постоянными коэффициентами.
а11=1, а12=-3, а22=9; (а12)2-а11[$183$]а22=9-9=0 [$8658$]
по классификации это уравнение параболического типа.
Характеристическое уравнение будет иметь вид
, или .Общий интеграл этого уравнения: y+3x=C.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Преобразуем производные к новым переменным, используя следующие формулы:
Получим
Подставив их в исходное 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=16, а12=8, а22=3; (а12)2-а11·а22=64-48=16>0 ⇒
по классификации это уравнение гиперболического типа.
Характеристическое уравнение будет иметь вид
Получим два уравнения
Общие интегралы этих уравнений: 4y-3x=C1,4y-x=C2.
Для приведения к канонической форме введем новые переменные:
Дифференцируем:
Пересчитаем производные 2-го порядка в новых переменных:
Подставив их в исходное 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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