Question

How do you determine the number of degrees of freedom of a differential equation?

Answer 1

As in the thing that determines the number of possible indepedent solutions, if I'm using the wrong name for it.

Answer 2

Yes, I am not used to hearing that terminology, bu from what I'm reading it just seems to be the number of constants present in the general solution
for instance p(x)y''+q(x)y'+r(x)y=f(x) will have two degrees of freedom as the complimentary solution will give us two constants that will act as parameters based on the initial conditions.

Answer 3

in general, an n-th order DE has n linearly independent solutions, so I suppose the order should be the same as the number of degrees of freedom... based on what I'm reading, which is not as detailed as I would like.

Answer 4

The 2 constants being a and b in ae^{bx}?

Answer 5

well in the example I gave we don't know what the solutions are, but no I mean\[y^{(n)}+y^{(n-1)}+...=f(x)\]will have solutions\[y_c+y_p=c_ny_n+c_{n-1}y_{n-1}+...+c_1y_1+yp\]the particular solution will not add any degrees of freedom, so we just look at the number of constants that come from the complimentary soln, which is based as I said on the order.
Better get s second opinion though, my book does not have great detail.

Answer 6

so counting the number of unknown constants c in the complimentary solution is literally your degrees of freedom

Answer 7

check out section 3.2 on this doc
http://math.uchicago.edu/~vipul/teaching-0910/153/introtodiffeq.pdf