# Substitution to convert first order ODE to homogenous

## 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

## 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

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
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.