Find all functions $g$ such that $$\nabla^2g = x^3+y^2+z$$
What I tried:
Basically $\nabla^2$ gradient of a divergence and we can write it as
$$\nabla^2 g = \frac{\partial^2g}{\partial x^2} + \frac{\partial^2g}{\partial y^2} + \frac{\partial^2 z}{\partial z^2} = x^3+y^2+z$$
How do I solve it? Help me, please.
First off, note that with
$g(x, y, z) = \dfrac{1}{20}x^5 + \dfrac{1}{12}y^4 + \dfrac{1}{6}z^3, \tag 1$
we have
$\nabla^2 g = \dfrac{\partial^2 g}{\partial x^2} + \dfrac{\partial^2 g}{\partial y^2} + \dfrac{\partial^2 g}{\partial z^2} = x^3 + y^2 + z; \tag 2$
now suppose $f(x, y, z)$ also satisfies (2), that is
$\nabla^2 f = x^3 + y^2 + z; \tag 3$
then subtracting (3) from (2) we find
$\nabla^2(g - f) = 0,\tag 4$
that is, $g - f$ is a harmonic function; likewise, with $g - f$ harmonic we have
$\nabla^2 (g - (g - f)) = \nabla^2 g - \nabla^2(g - f) = \nabla^2 g = x^3 + y^2 + z; \tag 5$
it follows that the set of all functions on $\Bbb R^3$ satisfying (2) are precisely those differing from $g(x, y, z)$ by a function harmonic on all of $\Bbb R^3$.