For the differential equation d y d x +i y=2 (x) , with initial condition y(0)= 3 2 . If y( )= a b +c i , for co-prime integers a and b , find a b c . [Note: i= -1 denotes the imaginary unit.]

Step-by-step solution

d y d x +i y=2 (x) (Linear D.E. with integrating factor e^ i x ) Hence, aligned y e^ i x = 2 x e^ i x d x & = 2 x( x+i x) d x & = ^2 x+i (x- 1 2 2 x )+c aligned Given aligned y(0) & = 3 2 c= 3 2 & y e^ i x = ^2 x+i (x- 1 2 2 x )+ 3 2 aligned Now put x= aligned y( ) e^ i & =0+i( -0)+ 3 2 y( )(-1) & =i + 3 2 aligned Hence, y( )=- 3 2 - i Therefore a b c=6