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Задать вопрос экспертам

Консультация № 187014

22.12.2012, 21:08

97.34 руб.

0 1 1

Образование

Здравствуйте! У меня возникли сложности с таким вопросом:

(5 пример из таблицы)

Заранее благодарю.

Обсуждение

давно

Профессор
323606
198

22.12.2012, 22:51

общий

это ответ

Здравствуйте, AND1!
Характеристическое уравнение [$955$]²+3[$955$]+3=0.
D=3²-12=-3; [$955$]_1,2=(-3[$177$][$8730$]3i)/2.
[$955$]_1,2=-3/2[$177$]([$8730$]3/2)i.
Общее решение однородного уравнения у_о=е^3х/2(С₁cos([$8730$]3x/2)+С₂sin([$8730$]3x/2)).
Частное решение неоднородного уравнения ищем в виде у^*=Acos2x+Bsin2x. Тогда (у^*)'=-2Asin2x+2Bcos2x,(у^*)''=-4Acos2x-4Bsin2x.
Подставляя в неоднородное уравнение, имеем:
-4Acos2x-4Bsin2x-6Asin2x+6Bcos2x+3Acos2x+3Bsin2x=cos2x,
-Acos2x-Вsin2x-6Asin2x+6Bcos2x=cos2x, [$8658$]
-А+6В=1,
-В-6А=0.
Отсюда А=-1/37, В=6/37.
у^*=(-1/37)cos2x+(6/37)sin2x.
Общее решение неоднородного уравнения у=у_о+у^*=е^3х/2(С₁cos([$8730$]3x/2)+С₂sin([$8730$]3x/2))-(1/37)cos2x+(6/37)sin2x.
Найдем частное решение, удовлетворяющее начальным условиям у(0)=у'(0)=0.
у(0)=С₁-1/37=0 [$8658$] С₁=1/37.
у'=(3/2)е^3х/2(С₁cos([$8730$]3x/2)+С₂sin([$8730$]3x/2))+е^3х/2(-([$8730$]3/2)С₁sin([$8730$]3x/2)+([$8730$]3/2)С₂cos([$8730$]3x/2))+(2/37)sin2x+(12/37)сos2x,
у'(0)=(3/2)С₁+([$8730$]3/2)С₂+12/37=0 [$8658$] С₂=2(-12/37-(3/2)С₁)/[$8730$]3=-27/(37[$8730$]3).
Искомое частное решние у=(1/37)(е^3х/2(cos([$8730$]3x/2)-(27/[$8730$]3)sin([$8730$]3x/2))-cos2x+6sin2x).

5

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