# Thread: Particular and Complementary Solutions

1. ## Particular and Complementary Solutions

$\displaystyle u_x-u_y+u=1$

$\displaystyle u_x-u_y+u=0$

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

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.

2. Originally Posted by dwsmith
$\displaystyle u_x-u_y+u=1$

$\displaystyle u_x-u_y+u=0$

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

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.
What about the method of undetermined coeffeints? The right hand side is a degree zero polynomial, so the particular solution must be a degree zero polynomial(a constant)

3. Originally Posted by TheEmptySet
What about the method of undetermined coeffeints? The right hand side is a degree zero polynomial, so the particular solution must be a degree zero polynomial(a constant)
I wasn't to sure about that either.

If we let $\displaystyle u_p=A$, then $\displaystyle u_{px}=0 \ u_{py}=0$

$\displaystyle 0-0+A=1$

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

So that is it?

4. yep, just plug it into the PDE to check that it satisfies the equation!