v=e^x\\z=x+Ce^{-x}=y^3\\x-y^3+Ce^{-x}=0\\y(1)=-1\\1+1+Ce^{-1}=>C=-2e\\x-y^3-2e^{1-x}=0\\\\\\1-3y^2y'-Ce^{-x}=0\\1-3y^2y'-y^3+x=0\\3y^2y'+y^3=x+1" alt="3y^2y'+y^3=x+1\\(y^3)'+y^3=x+1\\y^3=z\\z'+z=x+1\\z=uv;z'=u'v+v'u\\u'v+v'u+uv=x+1\\u'v+u(v'+v)=x+1\\\frac{dv}{dx}+v=0\\\int \frac{dv}{v}=-\int dx\\ln|v|=-x\\v=e^{-x}\\\frac{du}{dx}e^{-x}=x+1\\\int du=\int e^x(x+1)dx=e^x(x+1)-e^x+C=xe^x+C\\u=x+1;du=dx\\dv=e^xdx=>v=e^x\\z=x+Ce^{-x}=y^3\\x-y^3+Ce^{-x}=0\\y(1)=-1\\1+1+Ce^{-1}=>C=-2e\\x-y^3-2e^{1-x}=0\\\\\\1-3y^2y'-Ce^{-x}=0\\1-3y^2y'-y^3+x=0\\3y^2y'+y^3=x+1" align="absmiddle" class="latex-formula">