find the particular integral?

Question

find the complete integral of the PDE $$\frac{ \partial^2u }{\partial x^2} + 2 \frac{\partial^2u}{\partial x \partial y} + \frac{\partial^2u}{\partial y^2} = xe^{x+y}$$

My attempt : auxiliary equation is $m^2 +2m +1 =0$ ,that is $m = -1,-1$

so the complementary function will be $f_1 (y+x) + xf_2(y+x)$

Now the particular integral , $$P.I = \frac{xe^{x+y}}{(D+1)^2}$$

after that im not able to proceed further

Also i know that complete integral = C.F + P.I

any hints/solution will bne appreciated

thanks u

Answer 1

Since $(\partial_x^2+2\partial_x\partial_y+\partial_y^2) (x^n e^{x+y})=(2x^n+4nx^{n-1}+n(n-1)x^{n-2})e^{x+y}$, with special cases $(\partial_x^2+2\partial_x\partial_y+\partial_y^2)e^{x+y}=2e^{x+y}$ and $(\partial_x^2+2\partial_x\partial_y+\partial_y^2)(x e^{x+y})=(2x+4)e^{x+y}$, the particular integral is $(\frac{x}{2}-1)e^{x+y}$. (You'll want to double-check all my arithmetic.)