Solve the following system of simultaneous pde's (all derivatives are partial derivatives):
df/dx = 3x^2 .... (1)
df/dy = 5z^2 .... (2)
df/dz = 10yz .... (3)
From (1): f = x^3 + g(y, z).
From (2): dg/dy = 5z^2 => g = 5z^2 y + h(z).
Therefore f = x^3 + 5z^2 y + h(z).
From (3): 10 zy + dh/dz = 10yz => dh/dz = 0 => h = constant. Take h = 0.
Therefore f = x^3 + 5z^2 y.