Why Use Two Arbitrary Constants in Circle Parametrization for PDEs?

[kingwinner]

I am confused by the following example about solving quasilinear first order PDEs.

For the part I circled, the solution is just x^2 + y^2 = k where k is an arbitrary constant. To parametrize it in terms of t, can't we just put x = a cos(t), y = a sin(t) ? Here we only have one arbitrary constant a.
But in the example, they used a weird parametrization of a circle that includes TWO arbitary constants a and b. So my point is: why introduce another extra arbitrary constant when it is completely unnecessary to do so?

Can someone please explain why it is absolutely necessary to parametrize the circle in the way they do?

Any help is greatly appreciated! :)

[note: also under discussion in S.O.S. math cyberboard]

 

[LCKurtz]

If you look a couple of lines above what you circled you will see you need to solve the system:

x'(t) = y(t), y'(t) = -x(t)

So x''(t) = y'(t) = -x(t) giving x''(t) + x(t) = 0. This has the general solution:

x(t) = a cos(t) + b sin(t)
y(t) = -x'(t) = a sin(t) - b cos(t)

 

[kingwinner]

Hi LCKurtz,

OK, now I see why the circled part is correct by using your method. Your way actually makes more sense to me :)

But why is the author trying to combine the two equations dx/dt = y, dy/dt = -x ? How is this going to help us to solve the system?
Combining these two, we get dy/dx = -x/y, the general solution is just x^2 + y^2 = k where k is an arbitrary constant. To parametrize this general solution it in terms of t, just put x = a cos(t), y = a sin(t), right? This parametrization satisfies the equation x^2 + y^2 = k, so it must be a correct parametrization. What is wrong with this approach? Can you please point out where this line of logic fails?

Thank you!

 

[LCKurtz]

I don't know why your author eliminates the parameter. I wouldn't have, but then I also don't write PDE books or notes.

The problem generally with eliminating parameters is that you lose information. In your case, you have a system of 2 DE's and would expect a two parameter family of trajectories. That fact disappears when you eliminate the parameter.