Origin of the term "stiff" differential equation:

Analysis of the eigenvalues of the damped spring equation 

m y"  +  γ y' + k y = 0.

Setting (Subscript[v, 1]) = y, (Subscript[v, 2]) = y', we first obtain

((({{(Subscript[v, 1])}, {(Subscript[v, 2])}})))' = (({{(Subscript[v, 2])}, {-(γ/m) (Subscript[v, 2])  -  (k/m) (Subscript[v, 1])}})) , that is, v' = A v. Finding the eigenvalues of A, we obtain:

A  =  {{0,  1},  {-k/m , -γ/m} }

MatrixForm[A]

Eigenvalues[A]

This shows that, for a large spring constant, that is, for k large, corresponding to a "stiff" spring, but not so large that 4km > (γ^2), one eigenvalue is negative but near 0, while the other eigenvalue is negative but large.  (This, for example, could be a model of shock absorbers on an automobile, where γ is large and the absorber itself is relatively stiff, so k is large.)