Finding second order linear homogenous ODE from the fundamental set of solutions 
Find second order linear homogeneous ODE with constant coefficients if its fundamental set of solutions is {$e^{3t},te^{3t}$}.

Attempt: Had this question in my midterm. So, since the fundamental set of solutions is $$y=y_1+y_2=c_1e^{3t}+c_2te^{3t}$$
the characteristic equation of the second order ODE has only one root. I don't know what to do next. Help please.

The eigenvalues and eigenvectors for the coefficient matrix $A$ in the linear homogeneous system: 
$Y'= AY$ are $\lambda_{1} = 3$ with $v_1 =< a; b >$ and $\lambda_2 = 3$ with $v2 =< c; d >$
The fundamental form of the solution is:
$$ Y = c_1 e^{3t}v_1 + c_2t e^{3t}v_2$$
Take the second derivative, $Y''$ for the DEQ.
Your original system will be of the form:
$$y'' - 6y' + 9 = 0$$
to give you the double eigenvalue $\lambda_{1,2} = 3$. You can actually solve this to find the corresponding eigenvectors.
Regards
