I have to reduce my question to a homogeneous ODE.

Question

$$\frac{dy}{dx}= \frac{1-x-y+1}{y-x-5}$$

I've taken the substitutions $x=X+h$ and $y=Y+k$, but my answers appear to be wrong.

Answer 1

let $$x=X+k_1$$ $$y=Y+k_2$$ to cancel the constants $2$ and $-5$, it should be

$$k_1+k_2=-2\tag1$$ $$k_2-k_1=5\tag2$$ so $$k_1=-\frac{3}{2}$$ $$k_2=\frac{7}{2}$$

Answer 2

Substitute $$x=-1.5+u\\ y=3.5+v\\ \frac { dv }{ du } =\frac { -u-v }{ v-u } $$