What you have done is correct.
Neither of the two is linear or homogeneous. To see this, calculate L(cu) in either case. One ends up with an extra c and one cannot have a common factor of c.
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?