If u= ( x y ), x=e^t and y=t^2 , then t^6 ( d u d t )^2 e^ 2 t (t-2)^2=

Step-by-step solution

Given u= ( x y ), x=e^t, y=t^2 u= ( e^t t^2 ) Now, d u d t = ( e^t t^2 ) t^2 e^t-e^t 2 t (t^2 )^2 aligned & = ( e^t t^2 ) ( t-2 t^3 ) e^t & ( d u d t )^2= ^2 ( e^t t^2 ) (t-2)^2 t^6 e^ 2 t & t^6 ( d u d t )^2 (t-2)^2 e^ 2 t & = ^2 ( e^t t^2 )=1- ^2 ( e^t t^2 )=1-u^2 aligned