The purpose of this post is the same as it was this time last semester: to get my notes for my math test tomorrow down somewhere, just to help me remember it. This will be the format for the notes
Section number- what knowledge can be acquired by learning the section
How to do it, in Brian terms.
Here is the breakdown by sections:
7.1 Transforming a second order differential into a system of first order differential equations
Set x1 to u, aşa că u'= x'1
Then set x2 also to u', and then u''=x'2
Get the original equation in terms of x1 equal to x'2
7.2 Verify that a given vector satisfies a given equation
Learn basic linear algebra
7.3 Determining linear relationships between vectors
Give all vectors a constant in an equation similar to
C1x1+C2x2+C3x3=0
Write as an augmented matrix
Solve and pray they are all different constants
Part II Finding eigenvalues and eigenvectors
...Find the eigenvalues.
...Find the eigenvectors
7.4 Finding the Wronskian of a given vector, finding the interval of linear independence, find a system of equations that verify linear independence
You're given x1 and x2 vectors
Find W, and where it is zero, those values don't exist. That means that the interval of linear independence is everywhere that those values aren't.
The system of equations that verify linear independence is found by simply finding a vector that satisfies x'=Px
7.5 Find the general solution of a given system with different, real eigenvalues
Find the eigenvalues(these will be known as r1 and r2), then the eigenvectors (represented by ξ1 and ξ2)
Your solution is of the form x=c1ξ1er1t + c2ξ2er2t
...Yeah, it is taking too long to do the html code on this, and I won't do it otherwise... so I won't do it any more.
Bombachus
The person who realized you can pop corn deserves a Nobel Prize.