Solve Simple ODE: Prove y=Cx^k

[John O' Meara]

A mixture of two liquids X and Y, is boiling. At any time t there is present in the mixture a quantity x of one and y of the other. The ratio of the amounts of X and Y passing off at anyone time is proportional to the ratio of the quantities still in the liquid state. Prove that $$y=Cx^k \\$$, where C and k are constant.
I really don't know how to start this off, I have $$\frac{X}{Y} \propto \frac{x}{y} \\$$. What is the next step? Do I express X in terms of x and Y in terms of y? Thanks for helping.

 

[Dick]

Well, what is X/Y? X and Y are just labels. x and y are the quantities. Read the problem again and tell me what it says about the relation between x,y and x',y'.

 

[John O' Meara]

Does it say x'/y'=k*x/y, where x=x(t) e.t.c.

 

[Dick]

Sure. Now can you solve that ODE?

 

[John O' Meara]

I think I can.