Is This the Correct Approach to Solving Laplace's Equation?

[tesla93]

Homework Statement

Determine whether each of the following functions is a solution of Laplace’s
equation u_xx + u_yy = 0.

x^3 + 3xy^2

u_x=3x^2

u_xx=6x

u_y=6xy^2

u_yy=6x

6x+6x=12x and is therefore not a solution

Did I do that right? I'm just learning about this topic and it's a little hard to understand. Can anyone give any advice as to if I approached this correctly?

 

[xaos]

while your result is correct, your steps are not written down correctly.

 

[Telemachus]

Your derivatives are wrong. The procedure is correct, but your partial derivatives are wrong.

You have: $$f(x,y)=x^3+3xy^2$$
Then, the derivative with respect to x is:
$$f_x(x,y)=3x^2+3y^2$$

Everything else is okey. The result you found is fine because when you take the second derivative with respect to x the term involving y vanishes.