> with(LinearAlgebra):
> dsolve({# diff is the differentiation function
>         diff(y1(t), t) = 4*y1(t) - 2*y2(t),
>         diff(y2(t), t) = 1*y1(t) + 1*y2(t)
>        },
>        {# must list unknowns as functions in t
>         y1(t), y2(t)}
> );
> 

  {y1(t) = _C1 exp(3 t) + _C2 exp(2 t),

        y2(t) = 1/2 _C1 exp(3 t) + _C2 exp(2 t)}

# Note that the solution given in class is different because I used a
# different scalar multiple of the eigenvector corresponding to
# eigenvalue 3.  However, my y_2(t) is a linear combination of the above
# solution: 3*y1(t) - 2*y2(t).
> dsolve({# diff is the differentiation function
>         diff(y1(t), t) = 3*y1(t) + 4*y2(t),
>         diff(y2(t), t) = 3*y1(t) + 2*y2(t),
>         # initial conditions are additional equations
>         y1(0) = 6, y2(0) = 1},
>        {# must list unknowns as functions in t
>         y1(t), y2(t)}
> );
> 

   {y1(t) = 4 exp(6 t) + 2 exp(-t), y2(t) = 3 exp(6 t) - 2 exp(-t)}

> exp(2*Pi*I);

                                  1

> dsolve({diff(y1(t), t) =            y2(t),
>         diff(y2(t), t) =   -y1(t),
>         y1(0) = 1, y2(0) = 0
>        },{y1(t), y2(t)}
> );
> 
# 

                  {y1(t) = cos(t), y2(t) = -sin(t)}

> Eigenvalues(Matrix([[0,1],[-1,0]]));

                                 [I ]
                                 [  ]
                                 [-I]

>