It is known that partial differential fpartial differential

It is known that partial differential f/partial differential x = 2xy, and partial differential f/partial differential y = x^2 +1. Determine the form of f(x, y).

Solution

delf / delx = 2xy

Partially Integrating it with respect to x we get,

f = x²y + g(y) + C

Partially Differentiating it with respect to y we get

delf/dely = x² + g\'(y)

x² + 1 =  x² + g\'(y)

g\'(y) = 1

Integrating it with respect to y we get

g(y) = y + C

Thus, f(x,y) = x²y + y + C