$\displaystyle u_x-u_y+u=1$

$\displaystyle u_x-u_y+u=0$

$\displaystyle u(x,y)=C_4\exp{[x\lambda+y(\lambda+1)]}+u_p(x,y)$

What method is used to solve for $\displaystyle u_p(x,y)\mbox{?}$

I don't think the annihilator method can be used or I don't know how to use it with a PDE.