Question

Найти общее решение ду xy'=3xy-y

Answer 1

$xy'=3xy-y\\\\y'-3y=-\frac{y}{x}\\\\y=uv,\; y'=u'v+uv'\\\\u'v+uv'-3uv=-\frac{uv}{x}\\\\u'v+u(v'-3v)=-\frac{uv}{x}\\\\1)\; \; v'-3v=0\\\\\frac{dv}{v}=3dx\\\\ln|v|=3x,\; \; v=e^{3x}$

$2)\; \; u'e^{3x}=\frac{ue^{3x}}{x}\\\\\frac{du}{dx}=\frac{u}{x}\\\\\int \frac{du}{u}=\int \frac{dx}{x}\\\\ln|u|=ln|x|+ln|C|\\\\u=Cx\\\\3)\; y=uv=Cxe^{3x}$