The question defines $ u$ as a multivariable function: $ u = u(x,y)$.
Moreover, the question provides the partial derivative of $u$ in respect to $x$ as:
$$\frac{\partial ^2{u}}{\partial{x^2}} = 12xy$$
I am not sure about my final answer, which is:
$u(x,y) = 2x^3y+R(y)x+T(y)$
Is this right?
Based on @DanieleTampieri comment's, see bellow a more complete answer.
Since we have: $$\frac{\partial ^2{u}}{\partial{x^2}} = 12xy$$
Integrating in respect with x leads to:
$$\frac{\partial {u}}{\partial{x}} = 6x^2y +C_1(y)$$
In order to solve the PDE, we need to integrate in respect to x again:
$$u= 2x^3y + \int C_1(y)dx + C_2(y) $$
$$u= 2x^3y + C_1(y)\cdot x + C_2(y) $$
Let $C_1(y)= R(y) $ and $C_2(y)= T(y) $
The final answer is:
$ u(x,y) = 2x^3y+R(y)x+T(y) $
QED