Question

If x = t . log t, y = t^t, then show that `"dy"/"dx" - "y" = 0`

Answer 1

x = t . log t       

Differentiating both sides w.r.t. t

`dx/dt=td/dt(logt)+logtd/dt(t)`

= `txx1/t+logt(1)`

`dx/dt=1+logt`             ...(i)

y = t^t

Taking logarithm of both sides 

logy = logt^t

logy = t.log^t

Differentiating both sides w.r.t. t

`1/yxxdy/dt=td/dt(logt)+logtd/dt(t)`

= `txx1/t+logt(1)`

= 1 + logt

`dy/dt=y(1+logt)`     ...(ii)

`dy/dx=(dy/dt)/(dx/dt)`

= `(y(1+logt))/((1+logt))`     ...[From (i) & (ii)]

∴ `dy/dx=y`

∴ `dy/dx-y=0`