In order to talk about "linearly independent" solutions, you must be talking about linear differential equations.
The set of all solutions to a linear, homogeneous, differential equation, of order n, forms a vector space of dimension n so, yes, there are 3 "linearly independent" solutions to a third degree linear differential equation.
To prove this, consider the "fundamental" solutions, at some x= a. For an n degree equation, the fundamental solutions are solutions to the differential equation satisfying:
for y^(m)_n(a)= \delta_{mn}. That is, the nth derivative of y_n at x= a is 1 and all other derivatives are 0 there. (The "0" derivative is the value of the function.)
It is easy to show that those are "linearly independent" functions. Further if y(x) is any solution to the differential equation and [math]Y_n[/math] is the nth derivative of y at x= a, then [math]y(x)= \sum_{i= 0}^n Y_i y_i(x)[/math] so they also span the space.
