This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Notes for Day : . : Homogeneous Linear Systems A system of equations of the form: y y y = = = a (t)y + b (t)y + c (t)y + g (t) a (t)y + b (t)y + c (t)y + g (t) a (t)y + b (t)y + c (t)y + g (t) is said to be homogeneous if the functions g , g , etc. are all equal to zero. A homogeneous linear system can be written in matrix form as y = P(t)y. Example : Verifying a solution
Consider the system of equations: y y y e t e t = - y y y . Verify that the matrix y = is a solution to the system. Fundamental Sets, Solution Matrices, and the Wronskian
A fundamental set for a system of n linear equations is a set of n column vectors of functions, each of which is a solution to the system, and which are linearly independent. A fundamental set for the system in Example is: y = e t e t ,y = e t e t ,y = et . e general solution for Example could be written as: e t e t e t e t y=c But we can rewrite this as: +c +c et . y= e t e t e t e t et c c c . is called the solution matrix for the system. e solution matrix to a system of n et rst-order equations is an n n square matrix, whose columns form a fundamental set of solutions. e Wronskian of a solution matrix is the determinant of the solution matrix, and must be nonzero if the columns are linearly independent. e matrix e t e t e t e t Example a
Show that the solutions y , y , and y from Example are linearly independent. Example b
If we add the initial condition y( ) = to the system in Example , solve the initial value problem. ...
View Full Document