Let f be a differentiable function in the interval (0, ) such that f(1)=1 and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x0 . Then 2 f(2)+3 f(3) is equal to _______

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Let f be a differentiable function in the interval (0, ) such that f(1)=1 and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x>0 . Then 2 f(2)+3 f(3) is equal to _______

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aligned & _ t x t^2 f(x)-x^2 f(t) t-x =1 & _ t x 2 t . f(x)-x^2 f^ (x) 1 =1 & 2 x f(x)-x 2 f^ (x)=1 & d y d x - 2 x y= -1 x^2 & I.f. =e^ - 2 x d x = 1 x^2 & y x^2 = - 1 x^4 d x+C & y x^2 = 1 3 x^3 +C aligned Put f(1)=1 aligned & C= 2 3 & y= 1 3 x + 2 x^2 3 & y= 2 x^3+1 3 x & f(2)= 17 6 & f(3)= 55 9 & 2 f(2)+3 f(3)= 17 3 + 55 3 = 72 3 =24 aligned

Let f be a differentiable function in the interval (0, ) such that f(1)=1 and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x>0 . Then 2 f(2)+3 f(3) is equal to _______

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