Math Help - Linear ODE with const. coeff for a fundamental solution set

1. Linear ODE with const. coeff for a fundamental solution set

For the fundamental solution set S={ex,e2x,e3x} can we construct a linear ODE with constant coefficients?

I have verified that the solution set is linearly independent via wronskian. I have got the annihilators as (D-1),(D-2),(D-3). However after that I'm not sure how to proceed. What do I do to get the ODE?

Thanks

2. Re: Linear ODE with const. coeff for a fundamental solution set

I have progressed a little more:

From the annihilators, I get the characteristic equation to be: r3 - 6r2 + 11r - 6 = 0

Which gives me the linear ODE: y''' - 6y'' + 11y' - 6 = 0

Could anyone please confirm if this method is correct?