Diffeq quiz THREE (3)

What’s the first step that must be taken

Where does ln(x) not exist?

at 0

where does e^x not exist?

it exists everywehre

What steps must be taken?

You must use BOTH inital conditions. plug the y(x) condition into the y(x) equation. find y’(x) and plug the y’(x) condition in

what process is used to determine linear independence

Wronskian

So if the Wronskian is 0, what does this entail

If wronskian is 0, it is linearly dependent. if wronskian does NOT equal 0, it is linearly independent.

Wronskian for a 3×3

What steps must be taken to prove that the solutions are a fundamental step

1. plug into the general equation and prove the identity

2. confirm that it is linearly independent (W does NOT equal 0)

Wronskian for a 2×2 given y1,y2

How to plug in values to confirm it results in the identity

Characteristic equation for two real eigenvalues

Characteristic equation for repeated

Characteristic equation for complex

how to find the matrix eq?

When to use Cauchy euler

When the coefficients have variables

Cauchy euler’s Y(t) form

where a,b,c are the constants in the coefficients

what steps must be taken

1. convert to characteristic equation using the a,b,c equation

2. find the eigenvalues

3. find the general solution based on if the eigenvalues are real,complex,repeated

4. plug in t=ln(x)

   1. many of the e^lnx’s will turn into x’s!