Question

x-y-y'=-e^t

Answer 1

$y + y'=x+e^{x}$

$y = uv$

$uv + u'v + uv' = (u + u')v + uv'= x + e^{x}$

$u+u' = 0$

$u = e^{-x}$

$e^{-x}v' = x+e^{x}$

$v' = xe^{x} + e^{2x}$

$v = \int(xe^{x} + e^{2x})dx = \int xe^{x} dx + \frac{e^{2x}}{2} = xe^{x} - e^{x} + \frac{e^{2x}}{2} +c$

$y = uv = e^{-x}(xe^{x} - e^{x} + \frac{e^{2x}}{2} +c )$