Solution of the equation ^2 x d y d x -( 2 x) y= ^4 x,|x| 4 , where y ( 6 )= 3 3 8 , is given by

Step-by-step solution

Given differential equation can be written as d y d x - ( 2 x ^2 x ) y= ^2 x Here, P=- 2 x ^2 x = - 2 x 2 x ( 2 x+1 2 ) and Q= ^2 x IF =e^ P d x =e^ - 2 2 x 2 x( 2 x+1) d x Put 2 x=t -2 2 x d x=d t aligned If & =e^ 1 t ( 1 t+1 ) d t & =e^ ( 1 t - 1 t+1 ) d t & =e^ t- (t+1)] =e^ t t+1 & =e^ 2 x 2 x+1 = 2 x 2 x+1 aligned Now, solution is aligned y 2 x 2 x+1 & = 2 x 2 x+1 ^2 x d x+C & = 2 x 2 ^2 x ^2 x d x+C & = 1 2 2 x d x+C & = 1 2 2 x 2 +C aligned y 2 x 2 x+1 = 1 4 2 x+c ....(i) But y ( 6 )= 3 3 8 array ll 3 3 8 (2