MATH 20D Lecture 19

Section 9.7 - Solving Non-Homogeneous Linear Systems

A non-homogeneous linear system has the form , where .
The general solution to this equation is given by

where

• is the general solution to the corresponding homogeneous equation and
• is a particular solution to
  This is known, again, as the Superposition Principle.

Let $A$ be an $n\times n$ matrix and let ${\vec{x}}_1(t),{\vec{x}}_2(t),\dots ,{\vec{x}}_n(t)$ be $n$ linearly independent solutions to the homogeneous system ${\vec{x}}^{\prime }=A\vec{x}$. Then the fundamental matrix for the system ${\vec{x}}^{\prime }=A\vec{x}$ is given by

The general solution of can be expressed as

Variations of Parameters

Look for a particular solution of the form

Steps:

1. Write down , form the fundamental matrix , and find its inverse
2. Find
3. Find by integrating
4. Compute
5. Obtain the general solution

Undetermined Coefficients

Applicable to non-homogeneous linear system where

• is a matrix with constant entries
• All entries of are polynomials, exponentials, sines and cosines, or sums and products of the three
  Steps:

1. Write down and form the fundamental matrix
2. Make an initial guess for based on the form of the RHS
3. Solve for the coefficients in our guess for
4. Obtain the general solution

Issues w/ Undetermined Coefficient Method

1. When consists of more than one type of exponential
2. When there’s “conflict” b/w and