Solve the differential equations

Question

Solve the differential equations: Xy'–y=e^y'

Answer 1

Let differentiate both sides: y' + xy'' - y' = e^y'y'', or xy" - y''e^y' _{= 0; y''(x - e}^y'_{) = 0.}

_{These differential equation we can spleet on 2 equations: y'' = 0 and x - e}^y'_{= 0.}

_{From first equation we have y' = C. Let put this in differential equation: x·C - y = e}^C _{, y = Cx - e}^C_;

_{Now we solve second equation: x = e}^y'_{, y' = lnx; y = ∫lnxdx, y = xlnx - x + C. Thiks function satisfy given equation if C = 0.}

_{Answer: y = C x - e}^C _{and y = xlnx - x solutions of given equation}