I'm currently learning about homogeneous differential equations, and my material has this example
$$(x+y-2)dx + (x-y+4)dy = 0$$
This is not homogeneous, so to make it, we solve for the system of $$x + y = 2$$ $$x - y = -4$$ And get $x = -1$ and $y = 3$
Then we substitute, let $$x = \bar x - 1$$ $$y = \bar y + 3$$
Now, the original equation becomes $$(\bar x - 1 + (\bar y + 3 ) - 2)d(\bar x - 1) + (\bar x - 1 - (\bar y + 3) + 4)d(\bar y + 3) = 0$$
Simplifying the constants out, I get $$(\bar x + \bar y)d(\bar x - 1) + (\bar x - \bar y)d(\bar y + 3)$$
While the material I have goes straight from the unsimplified expression straight to $$(\bar x + \bar y)d\bar x + (\bar x - \bar y)d\bar y$$
And then solves the DE from there. I honestly am very confused with the substitutions for x and y where the differentials also gain the constants, but then in the final expression the constants are gone. My material goes straight from the unsimplified expression to the simplified one with $d\bar x$ instead of the one I got $d(\bar x - 1)$
Where is this simplification coming from? I can't continue learning the material until I understand why this happens, but I can't figure it out.
Thank you.
Recall that the derivative of a constant is zero. Hence $$d(\bar x - 1)=d(\bar x) - d(1)=d\bar x.$$