Methods that were used to solve the ODE

[NeutronStar]

Can anyone show me how to make this:

$$xy=c\left( y+\sqrt{y^2-x^2}\right)$$

Look like this:

$$y^2-cx=y\sqrt{ y^2-x^2}$$

These were given as the answers to an ODE problem.

I'm assuming that they are equivalent because they are presented in the answer as:

$y^2-cx=y\sqrt{ y^2-x^2}$, or equivalently $xy=c\left( y+\sqrt{y^2-x^2}\right)$

I got the second answer but I can't figure out how to algebraically manipulated it to make look like the first.

Any takers?

P.S. I'm not certain if it can even be done algebraically. It might have something to do with the methods that were used to solve the ODE. But if they are equivalent answers it should be possible to obtain one from the other using just algebra shouldn't it?

My algebra skills suck!

 

[Hurkyl]

It's true iff it can be proven algebraically. (Hurrah for abstract mathematics)

The method might become clear if you try solving both equations for c.

Better hint in white:


Try "rationalizing" the RHS -- multiply through by y - √(y² - x²). (and handle specially the case where this actually equals zero)

 

[NeutronStar]

Try "rationalizing" --

Ok, that was specifically the hint that I needed.

I guess I need to start doing some algebra problems everyday like morning calisthenics or something. It's hard to do algebra if you aren't doing it every day. I pretty much know how to do all the tricks, I just can never figure out which trick to pull out of the bag to achieve my final goals!

Thanks for the tip!

 

[Hurkyl]

Just FYI, I solved for c first. I figured rationalizing was the way to go, because one expression had the radical in the numerator, and the other on the denominator.

 

[NeutronStar]

Just FYI, I solved for c first. I figured rationalizing was the way to go, because one expression had the radical in the numerator, and the other on the denominator.

That little tid-bit of wisdom will definitely be useful to me in the future. To be perfectly honest, I'm so rusty with algebra that even that hint starting me in the face wouldn't have done much for me. But from now on when I see that situation I'll be thinking "rationalize!"

I'm just so rusty it's embarrassing.