I'm having trouble understanding what to do for this problem. The question I'm trying to answer is: Find 3 linearly independent solutions to the following differential equation, y^(3) + 3y'' + 3y' + y = 0. I really don't know how to even start this problem and what I'm really looking for. I think I need to try to find a characteristic equation, but I don't know if it will deal with complex numbers or not. So far the only thing I can think of for the characteristic equation is r^2(r+3) + 3r + 1 = 0. Below is what I have so far.
Can anyone tell me if I'm on the right track and if I've done these steps right so far? Also, what do I need to do next because I'm not sure how I know if I have 3 linearly independent solutions?y^(3) + 3y'' + 3y' + y = 0
r^2(r+3) + 3r + 1 = 0
r=0 of degree 2
r=-3 of degree 1
r=-1/3 of degree 1
y(x)=c1 + c2x + c3(e^(-3x)) + c4(e^((-1/3)x))