Let y=y(x) be the solution of the differential equation x ~d y ~d x +y x=4 x, x (0, ) . If y ( 2 )=0 , then y ( 6 ) is equal to

Step-by-step solution

aligned & x ~d y ~d x +y x=4 x & ~d y ~d x +y x= 4 x x & Here, P (x)= x, Q (x)= 4 x x & Integrating factor (I.F.) = e ^ P (x) d x & = e ^ oot x d x ,= e ^ | x| = x aligned array ll & y( I.F. )= Q ( I.F. ) d x+ c & y( x)= 4 x x x ~d x= C & y x=4 x ~d x+ C & y= 2 x^2+ C x & y ( 2 )=0 array ...(i) aligned & 2 ( 2 )^2+C ( 2 ) =0 & ^2 2 +C=0 & C= - ^2 2 aligned y= 2 x^2- ^2 2 x is the solution of the given differential equation. aligned y ( 6 ) & = 2 ( 6 )^2- ^2 2 ( 6 ) & = 2 ^2 36 - ^2 2 1 2 = -8 9 ^2 aligned