Recovering a Differential Equation From the Transfer Function of a Cascaded System
If the poles and zeros are cancelled then the analysis will be oblivious to the internal unstable modes.

To illustrate, I am going to use the system with the unstable pole at 2 that gets cancelled. A good set of differential equations to use is any state-space realization. 

I compute the state responses (using Mathematica)

Grid[{{tfm = 
    TransferFunctionModel[{{{s - 2}}, (s - 2) (s + 3) (s + 1)}, s],
   ssm = StateSpaceModel[tfm]}}, Background -&gt; LightBlue, Frame -&gt; All]
Grid[{Expand@StateResponse[ssm, 1, t]} // Transpose, 
 Alignment -&gt; Left, Spacings -&gt; {0, 2}, 
 Background -&gt; {None, {{LightOrange, White}}}, Frame -&gt; All]




The unstable mode \$e^{2 t}\$ appears in each of the state responses. However the output is \$-2 x_1+x_2\$ and the third state contributes nothing to the output. The amount of the unstable mode in the second state (\$1/15\$) is exactly twice that in the first state (\$1/30\$), and so the unstable mode does not appear in the output. 

This same analysis is readily obvious in a Jordan state-space realization of the system. Now the unstable mode appears only in the third state, and the third state simply does not appear in the output.

Grid[{{tfm = 
    TransferFunctionModel[{{{s - 2}}, (s - 2) (s + 3) (s + 1)}, s],
   jssm = Last@JordanModelDecomposition[ssm]}}, 
 Background -&gt; LightBlue, Frame -&gt; All]
Grid[{Expand@StateResponse[jssm, 1, t]} // Transpose, 
 Alignment -&gt; Left, Spacings -&gt; {0, 2}, 
 Background -&gt; {None, {{LightOrange, White}}}, Frame -&gt; All]




To do a complete analysis the poles and zeros should not be cancelled. They are prone to get canceled when we use transfer functions. This does not happen with a state-space realization unless it is explicitly done.

Both the state-space realization I used above give the same characteristic polynomial. I can get the minimal polynomial explicitly.

Factor@CharacteristicPolynomial[First@Normal@#, s] &amp; /@ {ssm, jssm, 
  MinimalStateSpaceModel@ssm}



  {-(-2 + s) (1 + s) (3 + s), -(-2 + s) (1 + s) (3 + s), (1 + s) (3 + 
      s)}