Undergrad Complex Analysis Harmonic functions

[Alvis]

Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic.

I tried using the Laplace Equation of Uxx+Uyy=0

I have:
du/dx=Ux
d^2u/dx^2=Uxx

du/dy=Uy
d^2u/dy^2=Uyy

dv/dx=cVx
d^2v/dx^2=cVxx

dv/dy=cVy
d^2v/dy^2=cVyy
I'm not really sure how to prove these are harmonic...am I missing a relationship?

 

[FactChecker]

Define w(x,y) = u(x,y) + cv(x,y) and calculate w_xx + w_yy. Basic properties of the partial derivative should give you the answer.

 

[Alvis]

Wow, that's a really good idea. I was trying to do the harmonic conjugate but was getting nowhere. Thank you!

 

[mathwonk]

do you understand what it means to say that differentiation is a linear operator?