The problem is 6y''-5y'+y=0, y(0)=4, y'(0)=0

I have found the two roots to be 1/2 and 1/3, so this gets us $\displaystyle y=c_1e^{1/2t}+c_2e^{1/3t}$

It's after this where I get kind of lost.

$\displaystyle y(0)=c_1+c_2=4$

Where do I go from here?