Answer to Question #315713 in Differential Equations for Davey

Question

dy/dx= y(xy^(5) − 1)

Answer 1

$y’=y(xy^5-1)=xy^6-y$

$z=y^{-5}$

$y’=-\frac15z^{-\frac65}z’$

$-\frac15z^{-\frac65}z’=x z^{-\frac65}-z^{-\frac15}$

$z’-5z=-5x$

$z=uv$

$u’v+uv’-5uv=-5x$

$u’v+u(v’-5v)=-5x$

$v’-5v=0$

$v=e^{5x}$

$u’e^{5x}=-5x$

$u=-\int 5xe^{-5x}dx=\frac15e^{-5x}(5x+1)+C$

$z=uv=x+\frac15+Ce^{5x}=y^{-5}$

$y=(x+\frac15+Ce^{5x})^{-\frac15}$

Answer: $y=(x+\frac15+Ce^{5x})^{-\frac15}$.