The function y(x) represented by x= t , y=a e^ t 2 +b e ^ t 2 , t ( - 2 , 2 ) satisfies the equation (1-x^2 ) y^ -x y^ = k y , then the value of k is

Step-by-step solution

aligned x & = t d x d t & = t y & =a e^ t 2 +b e^ t 2 & =e^ t 2 (a+b) aligned array ll & d y dt =( a + b ) e ^ t 2 2 & ~d y ~d x = d y dt d x dt = 2 ( a + b ) e ^ t 2 t array aligned & d^2 y d x^2 = d d t ( 2 (a+b) e^ t 2 t ) d t d x & = 2 (a+b) [ t (e^ t 2 2 )-e^ t 2 (- t) ( t)^2 ] 1 t aligned aligned & = 2 ( a + b ) [ e ^ t 2 ( 2 t + t) ^3 t ] & (1-x^2 ) y^ -x y^ aligned = (1- ^2 t ) 2 (a+b) [ e^ t 2 ( 2 t+ t) ^3 t ]- t [ 2 (a+b) e ^ t 2 t ] array r = ( ^2 t ) 2 (a+b) [ e^ t 2 ( 2 t+ t) ^3 t ] - t [ 2 (a+b) e^ i