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Задать вопрос экспертам

Консультация № 200935

23.05.2021, 22:31

0.00 руб.

0 2 1

Образование

Уважаемые эксперты! Пожалуйста, ответьте на вопрос: Решите уравнение:

Обсуждение

давно

Студент
405049
133

24.05.2021, 08:41

общий

Представим z как z=r*exp(j*[$966$])

Тогда уравнение запишется в виде

4*( r*exp(-j*[$966$]) )² - 4*r*exp(j*[$966$])*r + 3 = 0

или

4*r²*exp( -j*2*[$966$]) - 4*r²*exp( j*[$966$]) + 3 = 0

Используем выражение exp( j*[$966$]) = cos([$966$]) + j*sin([$966$]) и получим

4*r²*( cos(2*[$966$]) - j*sin(2*[$966$]) ) - 4*r²*( cos([$966$]) + j*sin([$966$]) ) +3 =0

или

( 4*r²*cos(2*[$966$]) - 4*r²*cos([$966$]) +3 ) -4*r²*j*( sin(2*[$966$]) - sin([$966$])) =0

Отсюда получаем систему уравнений

4*r²*cos(2*[$966$]) - 4*r²*cos([$966$]) +3 = 0

-4*r² *( sin(2*[$966$]) - sin([$966$]) )= 0

Сначала решаем последнее уравнение. Одно из решений r=0 не является решением первого уравнения, поэтому его отбрасываем. Поэтому решаем уравнение

sin(2*[$966$]) - sin([$966$]) = 0

Находим [$966$], подставляем в первое уравнение, находим r.

А затем, зная r, [$966$], а значит, и z=r*exp(j*[$966$]), вычисляем действительную и мнимые части z по формуле r*exp( j*[$966$]) = r*cos([$966$]) + j*sin([$966$])

и записываем окончательное решение уравнения.

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18345

28.05.2021, 08:17

общий

это ответ

Здравствуйте, mfti!

Имеем

[видно, что

]

Из уравнения (2) получим, что

При этом возможны следующие варианты:
1)

Тогда из уравнения (1) получим

или

Это утверждение ложно. Значит, при таких значениях

и

уравнение (1) не имеет решений;

2)

Тогда из уравнения (1) получим

Это уравнение не имеет решений в неотрицательных вещественных числах, каковым должно быть число

-- модуль комплексного числа. Значит, при таких значениях

и

уравнение (1) не имеет решений;

3)

Тогда из уравнения (1) получим

Значит, решением уравнения (1), а с ним и заданного уравнения, является число

4)

Тогда из уравнения (1) получим

Значит, решением уравнения (1), а с ним и заданного уравнения, является число

Получили, что единственным корнем заданного уравнения является число

Об авторе:
Facta loquuntur.

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