Separation of variables for nonhomogeneous PDE

Question

I need to solve the equation below using separation of variables. $$\frac{\partial f(x,y)}{\partial x} - \frac{\partial f(x,y)}{\partial y} = 2$$ The thing is, i've always done with $0$ after the equal sign. I'm really stuck with that $2$; when doing the separation I get $X'Y-XY'=2$ and can't separate X and Y after that.

Answer 1

$$\frac{\partial f(x,y)}{\partial x} - \frac{\partial f(x,y)}{\partial y} = 2$$ Substitute $f(x,y)=h(x,y)+2x$ $$\frac{\partial h(x,y)}{\partial x} - \frac{\partial h(x,y)}{\partial y} =0 $$

Answer 2

Let $(a,b)$ be some vector such that $a-b=2$. Then,

$$f_x-f_y=2=(1,-1) \cdot (a,b)$$

$$(1,-1) \cdot (f_x,f_y)=(1,-1) \cdot (a,b)$$

It follows that $(f_x-a,f_y-b)$ is orthogonal to $(1,-1)$. Let $g(x,y)=f(x,y)-ax-by$. Then $\nabla g$ is orthogonal to $(1,-1)$. So $\nabla g=\lambda(1,1)$ and $g(x,y)=\lambda (x+y)$. Hence $\lambda(x+y)+ax+by=f(x,y)$. Or with $b=2-a$:

$$f(x,y)=\lambda(x+y)+ax+(2-a)x$$

$$=\lambda x+ \lambda y+2 x$$