a detail discussion on solution of

a detail discussion on solution of linear differential equation, but what I want to say is that, when I look at that I have got two part, these are all differential equation with the differential coefficient, it
is very well known that you can express a solution in terms of a conventional exponential function.
It is known that for this equation, you can write the solution as the v can be written as v 1 e to the power of sigma 1 t plus v 2 e to the sigma 2 t and r can be expressed as r 1 e to the power of sigma 1 t plus r 2 e to the power of sigma sorry e to the power of sigma 2 t, where sigma 1 and sigma 2 are called roots of a characteristics equation, which is looking like this, but this is form by taking a determinant of that, actually I will just tell you, that I mean this is all very very standard stuff see if I if I write this equation again I will write this in a matrix form, that there will be easier.