Differential Equations Lecture 9 System of Linear Differential

Differential Equations Lecture 9 System of Linear Differential Equations fall semester Instructor: A. S. Brwa / MSc. In Structural Engineering College of Engineering / Ishik University

Ishik University Linear System A system of n linear equations in n variables can be expressed as a matrix equation Ax= b: Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 2

Ishik University Linear System Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 3

Ishik University Linear System Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 4

Ishik University Linear System A system of linear (algebraic) equations, Ax = b, could have zero, exactly one, or infinitely many solutions. Type of Linear Equation Number of Solutions Indication when Solving Conditional One Final line is x = a number. Identity Infinite; solution set {all real numbers} Final line is true, such as 0 = 0. None; solution set Final line is false, such as 0 = – 20. Contradiction Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 5

Ishik University Linear System Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 6

Ishik University Linear System Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 7

Ishik University Linear System Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 8

Ishik University Linear System Ax = b If the vector b on the right-hand side is the zero vector, then the system is called homogeneous. A homogeneous linear system always has a solution, namely the all-zero solution (that is, the origin). This solution is called the trivial solution of the system. Therefore, a homogeneous linear system Ax = 0 could have either exactly one solution, or infinitely many solutions. Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 9

Ishik University Linear System There is no other possibility, since it always has, at least, the trivial solution. If such a system has n equations and exactly the same number of unknowns, then the number of solution(s) the system has can be determined, without having to solve the system, by the determinant of its coefficient matrix: Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 10

Ishik University Linear System Theorem: If A is an n × n matrix, then the homogeneous linear system Ax = 0 has exactly one solution (the trivial solution) if and only if A is invertible (that is, it has a nonzero determinant). It will have infinitely many solutions (the trivial solution, plus infinitely many nonzero solutions) if A is not invertible (equivalently, has zero determinant). Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 11

Ishik University Eigenvalues and Eigenvectors Remember, the eigenvalues and eigenvectors are always exist for any given square matrix Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 12

Ishik University Eigenvalues and Eigenvectors Faculty of Engineering – Differential Equations – Lecture 9 – System of Linear Differential Equations 13