Difficult Parameterization technique

[^_^physicist]

Homework Statement

For a curve given by a^(2/3)=abs(x)^(2/3)+abs(y)^(2/3). Find a differentiable parameterization, all singular points, and determine their type.

Homework Equations

(Just common definations).

The Attempt at a Solution

Ok my thought was to find an arclength parameterization; however, the algebra is killing me. So far these are my steps, help would be greatly appreachited if you see a faster way.

abs(x)^2/3 + abs(y)^2/3=a^2/3 =>
1+abs(y^(2/3)/x^(2/3))=(a^(2/3))/(abs(x^(2/3))=>

[abs(y^(2/3)/x^(2/3))+1]^(3/2)=abs(a/x)

And then I get stuck. I have tried other methods...but they aren't much nicer.

Any algebra hints tips would be great...(I can't believe I have forgotten how to do these).

 

[AlephZero]

You just need to find any (differentiable) parameterization.

My first thoughts are:

Your equation is of the form

C = f_1(x) + f_2(y)

where f_1 and f_2 are both positive.

A well known equation that has a similar form is

1 = cos^2 t + sin^2 t

So I would try to invent a parameterization by putting those two ideas together, and then check that what I got was differentiable.