Suppose that Y is a n-dimensional vector. Consider the following linear differential equation system
$$dY/dt=AY$$
$$Y(0)=y_0$$
In which A is an n by n matrix.
My question is, can there be two different matrix A and B, such that $dY/dt=AY$ and $dY/dt=BY$ gives the same solution given the same initial condition? Or, I think that it is equivalent, can $exp(A*Y)$and $exp(B*Y)$ be the same when the exponential is defined as taylor series expansion?
If this problem is too complex, a link to papers would be enough.
Thank you!
The matrix differential equations $\dot{X} = AX$, $X(0) = I$ has solution $X(t) = e^{At}$.
If $A,B$ are such that $e^{At} = e^{Bt}$ for all $t$, then differentiating and setting $t=0$ gives $A=B$.