### Systems of differential linear equations

A system of differential linear equations is a system of the following form

Additionally we may have boundary conditions: . A solution is a vector of functions , which satisfies the system (if there are no boundary conditions given one should find all such functions.)

and boundary conditions (then we shall get one such vector).

Such a system can be written in a matrix form as:

If , the system is called homogeneous.

is thus an example of a system of differential linear equations and

is homogeneous. are exemplary boundary conditions for this system.

### Eigenvectors and solutions to a homogeneous system of differential linear equations

Notice that if is an eigenvector for an eigenvalue of matrix of a homogeneous system of differential linear equations, then is its solution. Indeed,

by the properties of derivative, but also by the definition of an eigenvector,

This means, that is matrix can be diagonalized (i.e. there exists a basis consisting of eigenvectors) then every solution of such system is a combination of basic solutions obtained this way.

### Solving homogeneous system of differential linear equations — an example

Let us solve

We are looking for eigenvalues of matrix

Its characteristic polynomial is . So we get eigenvalues and .

First we calculate a basis consisting of eigenvectors. For the eigenspace is described by , so it is spanned by . For it is described by , and thus spanned by .

Thus we get basic solutions: i , and each solution is their combination:

so

### Complex eigenvalues

The non-real eigenvalues of a real matrix go in pairs , . Their eigenvectors are also conjunct. This is the reason why in the general solution (after changing to ) it is possible to extract and , where are the constants of combination of basic solutions related to and respectively. Since we are looking for real solution we can choose ,

in such a way that and (their imaginary parts are negative of each other, and they have the same real parts). Thus, we can introduce real constants and .

Equivalently, instead of basic solutions and one can take and .

### Non-homogeneous systems of differential linear equations

The idea is similar to the idea of solving single non-homogeneous equation. We first solve the homogeneous version of the system and then change the constants to functions.

E.g. to solve

first we solve

which gives the solution

(see the previous section).

Changing the constants to functions we get:

so the derivatives are:

by substitution we get to:

which gives the following system of equations :

so the matrix of this system is

Thus,

Całkując dostajemy

So finally:

### What if the matrix is not diagonalizable?

Our considerations so far assume that matrix of the system is diagonalizable (at least in the complex numbers). What if it is not the case? How to get the missing basic solutions?

Notice first, that if is an eigenvector for eigenvalue , then if there exists such that , i.e. satisfying the system

then

is such that

thus it is also a basic solution!

If we still need more basic solutions, we can find such that , i.e. a vector satisfying

and then is such that

thus it is also a basic solution! An so on!

It can be proved that if is a -fold eigenvalue with only one dimensional eigenspace with basis , then there exist vectors described above and giving the missing basic solutions. These vectors are called Jordan’s vectors or the series of generalized eigenvectors.

E.g. let

We are looking for eigenvalues of

and there is only one: . Solving the system of equations

we get one-dimensional eigenspace spanned by . Thus we find any Jordan’s vector satisfying

e.g. . We need one more:

e.g. . Thus we get basic solutions

and

### Linear differential equations of higher order

It is a linear equation with derivatives of higher order, e.g.: .

Every such equation can be transformed into a system of equations of first order by stipulating , , etc. Then:

This is the system solved in the previous section. We get a solution for from it by looking at the first coordinate, so the general solution of the considered equation is