Differential equations help with substitution

If $x + 2y - 3 = v$

$2y = v - x + 3$

$y = \frac{1}{2}v - \frac{1}{2}x + \frac{3}{2}$

$\frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{1}{2}v - \frac{1}{2}x + \frac{3}{2}\right)$

$\frac{dy}{dx} = \frac{1}{2}\,\frac{d}{dx}(v) - \frac{1}{2}\,\frac{d}{dx}(x) + \frac{d}{dx}\left(\frac{3}{2}\right)$

$\frac{dy}{dx} = \frac{1}{2}\,\frac{d}{dv}(v)\,\frac{dv}{dx} - \frac{1}{2}$

$\frac{dy}{dx} = \frac{1}{2}\,\frac{dv}{dx} - \frac{1}{2}$ .

So substituting into your DE

$\frac{dy}{dx} = \frac{2}{x + 2y - 3}$

$\frac{1}{2}\,\frac{dv}{dx} - \frac{1}{2} = \frac{2}{v}$

$\frac{dv}{dx} - 1 = \frac{4}{v}$

$\frac{dv}{dx} = \frac{4}{v} + 1$

$\frac{dv}{dx} = \frac{v + 4}{v}$

$\frac{dx}{dv} = \frac{v}{v + 4}$

$\frac{dx}{dv} = \frac{v + 4}{v + 4} - \frac{4}{v + 4}$

$\frac{dx}{dv} = 1 - \frac{4}{v + 4}$

$x = \int{1 - \frac{4}{v + 4}\,dv}$

$x = v - 4\ln{|v + 4|} + C$

$x = x + 2y - 3 - 4\ln{|x + 2y - 3 + 4|} + C$

$x = x + 2y - 3 - 4\ln{|x + 2y + 1|} + C$

$4\ln{|x+2y+1|} = 2y - 3 + C$

$\ln{|x+2y+1|}= \frac{1}{2}y - \frac{3}{4} + C$

$|x + 2y + 1| = e^{\frac{1}{2}y - \frac{3}{4} + C}$

$|x + 2y + 1| = e^{-\frac{3}{4} + C}e^{\frac{1}{2}y}$

$x + 2y + 1 = \pm e^{-\frac{3}{4} + C}e^{\frac{1}{2}y}$

$x + 2y + 1 = Ae^{\frac{1}{2}y}$

$x = Ae^{\frac{1}{2}y} - 2y - 1$ .