Phase Plane Method - Example of A Linear System - Solving Using Eigenvalues

Solving Using Eigenvalues

More commonly they are solved with the coefficients of the right hand side written in matrix form using eigenvalues λ, given by the determinant:

and eigenvectors:

The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients. If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants c1, c2, ... cn.

The general solution is:

where λ1 and λ2 are the eigenvalues, and (k1, k2), (k3, k4) are the basic eigenvectors. The constants c1 and c2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.

The above determinant leads to the characteristic polynomial:

which is just a quadratic equation of the form:

where;

("tr" denotes trace) and

The explicit solution of the eigenvalues are then given by the quadratic formula:

where

Read more about this topic:  Phase Plane Method, Example of A Linear System

Famous quotes containing the word solving:

    If we parents accept that problems are an essential part of life’s challenges, rather than reacting to every problem as if something has gone wrong with universe that’s supposed to be perfect, we can demonstrate serenity and confidence in problem solving for our kids....By telling them that we know they have a problem and we know they can solve it, we can pass on a realistic attitude as well as empower our children with self-confidence and a sense of their own worth.
    Barbara Coloroso (20th century)