A linear ordinary differential equation is a differential equation in the form
Where is an unknown function and
The order, which is equal to , of the differential equation and it’s the highest order derivative seen in the equation.
For a linear ODE to be homogenous it simply means that .
The Characteristic Polynomial of the equation is .
the vector space of real functions n times differentiable.
We can also notice that differentiation is a Linear Operator in .
Since the equation is a linear combination of after varying amounts of differentiation - a linear operator - we can write the whole equation as
Where is a linear operator and is a polynomial of operator the differentiation operator. This means we can represent and as a Matrix. We will be using notation.
Given the solution space is the null space of which is equivalent to the Eigen space of at (As we have the familiar form of the equation )(Also since we are over a vector space of functions the eigen vectors are sometimes called eigen functions).
An eigen function of the base operator is clearly as it satisfies .
we have then
So we know that is equivalent to substituting this into we get this is the characteristic polynomial. If is a solution to the equation it is an eigen value of and we know that the corresponding eigen vector is
Reduction to a System of Degree 1 ODEs
For the homogenous case we define two vectors and and rearranging the equation to get .
Using this we can write the matrix equation as