notes - Solving Non-homogeneous Systems of Equations These...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solving Non-homogeneous Systems of Equations These notes describe how to solve (i.e., find the general solution of) a differential equation of the form d x dt = A ( t ) x + g ( t ) , when A is a 2 2 matrix. Note carefully that we assume that the d x dt term has coefficient 1. There are three methods: #1: Change of basis. #2: Undetermined coefficients. #3: Variation of parameters. Methods #1 and #2 can only be used if A ( t ) is independent of t , i.e., a constant matrix. Moreover, method #2 requires g to be of specific form: it can only contains terms that involve positive powers of t , e at , cos( at ), and sin( at ). Method #3 can always be used. It is recommended that you use method #3 because it always applies and is usually shorter than method #1. When method #2 can be used, it involves easier math (if you find solving vector equations easier than integration) but it can be tricky to set up for certain equations. Method #1: change of basis The idea here is to find a matrix P such that P- 1 AP is of a certain form and then introducing a new variable y = P- 1 x . The equation for x leads to an equation for y , namely d y dt = P- 1 AP y + P- 1 g ( t ) . This equation will be easier to solve than the original and once you solve for y , you then simply use the fact that x = P y to get the solution for x . There are three cases depending on A (for details see the notes on exponentials of ma- trices)....
View Full Document

Page1 / 4

notes - Solving Non-homogeneous Systems of Equations These...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online