Question

решить уравнение y'+y=x​

Answer 1

Ответ:

$y'+y=x\\\\y=uv\ ,\ \ y'=u'v+uv'\\\\u'v+uv'+uv=x\\\\u'v+u(v'+v)=x\\\\a)\ \ v'+v=0\ \ ,\ \ \int \dfrac{dv}{v}=-\int dx\ \ ,\ \ ln|v|=-x\ \ ,\ \ v=e^{-x}\\\\b)\ \ u'v=x\ \ ,\ \ \dfrac{du}{dx}\cdot e^{-x}=x\ \ ,\ \ \int du=\int x\cdot e^{x}\, dx\\\\\int x\cdot e^{x}\, dx=\Big[\ u=x\ ,\ du=dx\ ,\ dv=e^{x}\, dx\ ,\ v=e^{x}\ \Big]=\\\\=x\cdot e^{x}-\int e^{x}\, dx=x\cdot e^{x}-e^{x}+C\\\\u=e^{x}\cdot (x-1)+C\\\\c)\ \ y=e^{-x}\cdot \Big(e^{x}\cdot (x-1)+C\Big)\\\\y=(x-1)+Ce^{-x}$