University Mathematics Handbook (2015)

XI. Ordinary Differential Equations, or ODE

Chapter 4. Systems of First-Order Linear Equations

4.1  Definition

a.  The system

(*)

is a non-homogeneous linear ODE system.

b.  If , then the system is homogeneous.

c.  if functions and , are continuous on , and , then system (*) has a unique solution holding the initial conditions.

(**)

d. Vector form of a first-order ODE system

where

4.2  Homogeneous System of ODE

a.  Set of Vectors is a basis of a solution space, if they are LI solutions of system (*) on . In this case, any other solution is their linear combination.

b.  The determinant is called Wronskian.

c.  If are -solutions of (*) with continuous coefficients on , then, in this interval, the Wronskian is identically equal to zero or never vanishes.

4.3  Homogeneous System of First-Order Linear Equations with Constant Coefficients

Let be linear ODE when is constant matrix.

a.  If matrix has different eigenvalues and eigenvectors corresponding them, then

is a basis of the solution space.

b.  If eigenvalue of matrix has algebraic multiplicity but its geometric multiplicity is smaller than the algebraic one, we construct linearly independent vectors in the following way

,

Vectors are a set of LI eigenvectors, corresponding to and

are k LI solutions of the system corresponding to .

4.4  Non-homogeneous System of First-Order Linear Equations

(*)

Let

be an matrix, where the columns are solutions of the corresponding homogeneous system. The general solution of the homogeneous system is , when is a constant vector.

Varying the parameter, we look for a particular solution of system (*) in the form of . Substituting in (*), we get equation system , the solutions of which are . Therefore, the final solution is

when is an arbitrary vector.