`If f(x) =x + int_(0)^(1) (xy^(2)+x^(2)y)(f(y)) dy, "find" f(x)` if x and y are independent.

If f (x/y)= f(x)/f(y) , AA y, f (y)!=0 and f' (1) = 2 , find f(x) .

If f (x/y)= f(x)/f(y) , AA y, f (y)!=0 and f' (1) = 2 , find f(x) .

let f (x) = int _(0) ^(x) e ^(x-y) f'(y) dy - (x ^(2) -x+1)e ^(x) Find the number of roots of the equation f (x) =0.

If f(x+y) = f(x) + f(y) + |x|y+xy^(2),AA x, y in R and f'(0) = 0 , then

If f(x) + f(y) = f(xy-sqrt(1-x^(2))sqrt(1-y^(2))), f(0) = (pi)/(2) and f is differentiable in (-1, 1). Then |lim_(x rarr 0)(2f(x)-pi)/(x)|=

Let f((x+y)/(2))=(f(x)+f(y))/(2) and f(0)=b. Find f''(x) (where y is independent of x), when f(x) is differentiable.

Find f(x) if it satisfies the relation f(x) = e^(x) + int_(0)^(1) (x+ye^(x))f(y) dy .

Find f(x, y) if f(x + 2y, 2x + y) = 2xy.

If y=f(x) is the solution of equation ydx+dy=-e^(x)y^(2) dy, f(0)=1 and area bounded by the curve y=f(x), y=e^(x) and x=1 is A, then

A function f, continuous on the positive real axis, has the property that for all choices of x gt 0 and y gt 0, the integral int_(x)^(xy)f(t)dt is independent of x (and therefore depends only on y). If f(2) = 2, then int_(1)^(e)f(t)dt is equal to