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## Homework Statement

Make the following change of variables:

[tex]x = r \cos \theta[/tex]

[tex]y = r \sin \theta[/tex]

and integrate the following equation:

[tex](xy'-y)^2 = a(1+y'^2)\sqrt{x^2+y^2}[/tex]

## The Attempt at a Solution

First it's worth noting that the equation [tex]x^2+y^2=a^2[/tex] (even without changing variables) is solution for the above differential equation.

Now, making the substitution of variables, I'm able to reduce the equation down to:

[tex]\left(\frac{dr}{d\theta}\right)^2 = \frac{r^2(r-a)}{a}[/tex]

Looking at this equation, we see that if we take either r=0, or else r=a, then we get:

[tex]\frac{dr}{d\theta} = 0[/tex]

[tex]r(\theta) = K[/tex]

which, indeed, fitting w/ the "boundary condition" r=0 or r=a, gives us a consistent solution.

My question is: is this the "correct" way to solve the equation? Just by looking at the equation and making a "guess" that happens to work? Or is there a more "formal" way to take the equation:

[tex]\frac{dr}{r\sqrt{r-a}} = \pm\frac{d\theta}{a}[/tex]

and "derive" the "correct" solution?