Предмет: Алгебра, автор: МяяУкОло

Найти общее решение неоднородного линейного дифференциального уравнения, используя метод подбора коэффициентов частного решения (метод неопределенных коэффициентов)

y"+y=2cos7x+3sin7x

Ответы

Автор ответа: NNNLLL54

0

Ответ:

$y''+y=2cos7x+3sin7x\\\\a)\ \ k^2+1=0\ \ ,\ \ \ k^2=-1\ \ ,\ \ \ k_1=-i\ ,\ k_2=i\\\\y_{oo}=C_1\, cosx+C_2\, sinx\\\\b)\ \ f(x)=e^{0x}\cdot (2cos7x+3sin7x)\ \ ,\ \ \ 0+7i\ne 0+i\ \ \to \ \ r=0\\\\\widetilde {y}=Acos7x+Bsin7x\\\\\widetilde {y}'=-7Asin7x+7Bcos7x\\\\\widetilde {y}''=-49Acos7x-49Bsin7x\\\\\widetilde {y}''+\widetilde {y}=-48Acos7x-48Bsin7x=2cos7x+3sin7x\\\\-48A=2\ \ ,\ \ A=-\dfrac{1}{24}\\\\-48B=3\ \ ,\ \ B=-\dfrac{1}{16}\\\\\widetilde {y}=-\dfrac{1}{24}\, cos7x-\dfrac{1}{16}\, sin7x$

$c)\ \ y=y_{oo}+\widetilde {y}=C_1cosx+C_2sinx-\dfrac{1}{24}\, cos7x -\dfrac{1}{16}\, sin7x$

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