is the following linear and if linear state if linear inhomogeneous, or linear homogenous

L(U)=U_x+UU_y

L(U)=Y^2CosxU_x + U_(yxy)

I am confused do I first have to find L(u+v) and then L(CU)

Then do I have this correct

L(U)=U_x+xU_y

L(u+v)=(U+V)_x+x(U+V)_Y

=U+V_x + xU_y+xV_y

= L(u) + L(v)

L(CU)= (cu)_x +x(cu)_Y

= cu_x + xcu_y

= c( u_x + xu_y)

= cL(u)

Then this is linear right, do I have to do the same thing to the other two, then how do I tell if it is homogenous or inhomogenious?