Substitution to convert first order ODE to homogenous

[stfz]

Homework Statement

Use the substitution $x=X+h$ and $y=Y+k$ to transform the equation
$\frac{dy}{dx}=\frac{2x+y-3}{x-2y+1}$ to the homogenous equation
$\frac{dY}{dX}=\frac{2X+Y}{X-2Y}$
Find h and k and then solve the given equation

Homework Equations

The Attempt at a Solution

If I simply make the substitution into the equation, I get a homogenous equation which I can solve using y=vx substitution. But what I need help understanding is how the $\frac{dy}{dx}$ becomes $\frac{dY}{dX}$ after simply substituting into the LHS?
Is some proof or method of doing this so that I can turn dy/dx into dY/dX and vice versa? The chain rule doesn't help, as I cannot relate X and Y

 

[HallsofIvy]

First $$\frac{dY}{dx}= \frac{d(y- k)}{dx}= \frac{dy}{dx}$$

Now use the chain rule. $$\frac{dY}{dX}= \frac{dY}{dx}\frac{dx}{dX}= \frac{dy}{dx}(1)$$

The real point is that both x= X+ h and y= Y+ k are linear with slope 1.