Uniqueness of DE

Question

Question:

Suppose we have three roots of a cubic auxiliary equation from which two of them are repeated. How to determine whether the DE(Differential Equation) is unique or not?

Note that I am not talking about a uniqueness of the solutionn of a DE.

My idea:

I have concluded that the answer will not be a unique because two roots are repeated. But I don't understand how to determine in a proper manner. Say for example, if $m_1=4$ and $m_2=m_3=5$ are the roots of a cubic auxiliary equation.

Answer 1

If you speak about the "auxiliary equation", then the task is about a linear differential equation of order 3 with constant coefficients.

Any linear DE with continuous coefficients has solutions over the domain of the coefficients, and all solutions are unique. The multiplicity of the characteristic roots has no influence on that.

In your example, you get a solution family $$ y(t)=Ae^{4t}+Be^{5t}+Cte^{5t} $$ and any initial condition fixing values for $y(t_0)$, $y'(t_0)$, $y''(t_0)$ can be uniquely transformed into coefficients $A,B,C$.