Nonhomogenous Linear System

Solution Form

where is a particular solution to the original (nonhomogenous system) where are general solutions to the homogenous system of

Suppose that the solutions of the homogenous system are

The following matrix is called a fundamental matrix

Then

Let Then Integrate both sides to get Then is a particular solution

Example

Find first, start with eigenstuff.

Find fundamental matrix

Set , which implies Evaluate

Recall ![[Linear Algebra#2-times-2-inverse-shortcut| Inverse Shortcut]]

Integrate :

Acquire

Final answer