- #1
psholtz
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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?