Solve: dy/dx = (y - x + 1)/(y + x + 5)
rewrite as,
dy/dx = {(y + 3) - (x + 2)} / {(y + 3) + (x + 2)}
Let v = y + 3, w = x + 2
Differentiating,
dv/dx = dy/dx, dw/dx = 1
Dividing the two differential coefficients by each other,
(dv/dx) / (dw/dx) = dv/dw = (dy/dx) / (1) = dy/dx = (v - w)/(v + w)
i.e. dv/dw = (v - w)/(v + w) = {(v/w) - 1}/{(v/w) + 1} = (r - 1)/(r + 1)
so, dv/dw = (r - 1)/( r + 1), where r = v/w
or, v = rw giving dv/dw = r + w.dr/dw
substituting back in, (for dv/dw)
r + w.dr/dw = (r - 1)/(r + 1)
w.dr/dw = (r - 1)/(r + 1) - r = (r - 1)/(r + 1) - r(r - 1)/(r + 1) = {(r - 1 - r^2 - r) / (r + 1)}
w.dr/dw = -(1 + r^2)/(1 + r)
cross-multiplying and integrating both sides wrt w,
int (1 + r)/( 1 + r^2) dr = int (-1/w) dw
int 1/(1 + r^2) dr + int r/(1 + r^2) dr = -int (1/w) dw
atan(r) + (1/2)ln(1 + r^2) = -ln(w) + C
atan(v/w) + (1/2)ln(1 + (v/w)^2) + ln(w) = C
atan(v/w) + (1/2)ln([w^2 + v^2]/[w^2]) + ln(w) = C
Solution: atan([y + 3]/[x + 2]) + (1/2)ln({[x + 2]^2 + [y + 3]^2}/{[x + 2]^2}) + ln(x + 2) = C