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, ... c_n.

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