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Задать вопрос экспертам

Консультация № 179203

22.06.2010, 23:01

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Образование

Уважаемые эксперты помогите решить дифференцтальное уравнение по формуле: y(x)=˥y(x)+z(x)
Уравнение: y''-4y'+8y= (e^(2x)) +((sinx)^2)

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

23.06.2010, 10:20

общий

это ответ

Здравствуйте, Гапов Иван Сергеевич.
1) Сначала решаем однородное уравнение y''-4y'+8y=0. Для этого составляем характеристическое уравнение
[$955$]²-4[$955$]+8=0.
Оно имеет корни [$955$]=2[$177$]2i. Отсюда следует, что общее решение однородного уравнения
y=C₁e^2xsin2x+C₂e^2xcos2x

2) Правую часть можно представить в виде
f(x)=e^2x+(1/2)-(1/2)cos2x
Далее подбираем частные решения для каждого слагаемого

3) f₁(x)=e^2x
Частное решение ищем в виде y=Ae^2x. Подставляя в уравнение, находим
4Ae^2x-8Ae^2x+8Ae^2x=e^2x ---> A=1/4

4) f₂(x)=1/2. Частное решение ищем в виде y=A. Подставляя в уравнение, находим
0-0+8A=1/2 ---> A=1/16

5) f₃(x)=-(1/2)cos2x. Частное решение ищем в виде y=Asin2x+Bcos2x. Подставляя в уравнение, находим
(-4Acos2x-4Bsin2x)-4(2Acos2x-2Bsin2x)+8(Asin2x+Bcos2x)=-(1/2)cos2x
Приравнивая коэффициенты при подобных членах получаем систему
4A+8B=0
-8A+4B=-1/2
Решая систему, находим A=1/20, B=-1/40

Ответ: y=(1/4)e^2x+(1/16)+(1/20)sin2x-(1/40)cos2x+C₁e^2xsin2x+C₂e^2xcos2x

5

Спасибо!

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