Using the scalar equation as a quide, we assume that the vector equation (2) has a solution of the form

where is a constant vector.

Substituting the solution (5) in left and right side of equation (2) yields

Evidently, the vector equation (2) is satisfied only, if

i.e.

Therefore, if is an eigenvalue of A and is a corresponding eigenvector, then is a solution to ( gives a trivial solution). Obviously, the solutions

associated with eigenvalues , ..., and eigenvectors , ..., , are linearily independent. It is easy to verify that any linear combination of , ..., is also a solution. The general solution to the vector equation (2) is given by

Substituting (10) in (11) yields

The initial value problem consists in finding a solution to (2) that satisfies an initial condition

It follows from (12), (13) that

Determining the integration constants from linear algebraic system (14) and inserting in (12) one obtains the solution to initial value problem (1), (13).

General solution algorithm using linear algebra software (MATLAB, MAPLE,...) is following:

1. find the eigenvalues and eigenvectors

2. compose the general solution

3. determine the integration constants

4. compose the solution to the initial value problem.

Note: Steps 3-4 must be filled only for initial value problems.

Example 1: Solve the initial value problem

In matrix notation the system (15) reads

The characteristic polynomial of the matrix A is

Hence, the eigenvalues are and . Substituting the eigenvalues and in (9), we can determine corresponding eigenvectors

Now, the general solution of equation (15) can be presented as

The initial conditions (16) in matrix notation take the form

It follows from (20), (21) that

Solving system (22) yields: and . Inserting and into general solution (20) gives the solution to the initial value problem (15), (16) as

Example 2: Find the general solution to the linear system of differential equations

The matrix of system (24)

has the eigenvalues and . So, is an eigenvalue of multiplicity 2. Generally, if is an eigenvalue of A of multiplicity k, and the rank of the matrix is equal to n-k, then we can find k linearily independent eigenvectors of A associated with the eigenvalue . For given A with (25)

Therefore, we can find two linearily independent eigenvectors associated with using relation (9)

Computing the eigenvector associated with the eigenvalue

we can determine the general solution to system (24) as

Exercises

1. Solve the initial value problem

2. Find the general solution of the following linear system of differential equations:

3. Solve two point boundary value problem with differential equations (30) and boundary conditions x₁(1)=1 and x₂(0)=0.