Let `y =f (x)` satisfies the differential equation `xy (1+y) dx =dy . If f (0)=1 and f (2) =(e ^(2))/(k-e ^(2)),` then find the value of k.

If K is constant such that xy + k = e ^(((x-1)^2)/(2)) satisfies the differential equation x . ( dy )/( dx) = (( x^2 - x-1) y+ (x-1) and y (1) =0 then find the value of K .

Let y=f(x) satisfy the differential equation (dy)/(dx)=(x+y)/(x),y(1)=1, then y((1)/(e)) is equal

If y=f(x) satisfies the differential equation (dy)/(dx)+(2x)/(1+x^(2))y=(3x^(2))/(1+x^(2)) where f(1)=1 , then f(2) is equal to

If y=f(x) satisfy differential equation ( dy )/(dx)-y=e^(x) with f(0)=1 then value of f'(0) is

If y=f (x) satisfy the differential equation (dy)/(dx) + y/x =x ^(2),f (1)=1, then value of f (3) equals: