Let f:(0, ) R be a function which is differentiable at all points of its domain and satisfies the condition x^2 f^ (x)=2 x f(x)+3 , with f(1)=4 . Then 2 f(2) is equal to :

Step-by-step solution

aligned & x^2 f^ (x)-2 x f(x)=3 & ( x^2 f^ (x)-2 x f(x) (x^2 )^2 )= 3 (x^2 )^2 & d d x ( f(x) x^2 )= 3 x^4 aligned Integrating both sides aligned & f(x) x^2 =- 1 x^3 +C & f(x)=- 1 x +C x^2 & put x=1 & 4=-1+C C=5 & f(x)=- 1 x +5 x^2 & Now 2 f(2)=2 [- 1 2 +5 2^2 ] & =39 aligned