Differential substitution

consider:
t*dx/dt = x(1 + log(x/t)), x/t > 0

substitute u= x/t and show that the diff eq reduces to:
t(du/dt) = u*log(u)

please help

t dx/dt = x [ 1 + log(x/t)] ---------(1)

let u = x/t

=> x = ut
dx/dt = u + t du/dt

substitute in eqn (1)

t(u + t du/dt) = ut (1 + log(u))

ut + t^2 du/dt = ut _+ ut log(u)

t^2 du/dt = ut log(u)

divide by t

t du/dt = u log(u)