Eigenfunctions for Systems of LODEs

class: center, middle, inverse, title-slide

.title[
# Eigenfunctions for Systems of LODEs
]
.subtitle[
## Session 07
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-29
]

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# Goals

* Use eigenvalues and eigenvectors to solve systems of ODEs. <br/><br/>
* Use the Wronskian to test for linear independence. <br/><br/>

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# Conventions for Systems of ODEs

> In what follows we will deal with systems of two equations with two unknown functions whose ODEs are linear with constant coefficients. This is described in the general form `$$\begin{array}{c} x'= a_{11}x+a_{22}y+f_1(t) \\ y'=a_{21}x+a_{22}y+f_2(t) \end{array}$$` where `$a_{ij}$` are constants and `$f_1(t)$` and `$f_2(t)$` are functions of `$t.$`

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# Solving for Homogeneous systems

> Given the system `$$X'=AX$$` consider the matrix `$A.$` Suppose that `$(\lambda,v)$` is an eigenvalue-eigenvector pair of `$A$` and assume that `$\lambda$` is real. Let `$F=e^{\lambda t}v$` be a vector of functions, then the vector `$F$` is a solution vector of the system.

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#### Example:

> Consider the system `$$X'=AX$$` where `$$A=\left(\begin{array}{cc} 3 & 1 \\ 0 & -3 \end{array}\right).$$` Find the eigenvalues and eigenvectors of `$A$` and construct the general solution.

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#### Example:

> Consider the system `$$X'=\left(\begin{array}{cc} 1 & 2 \\ -2 & -3 \end{array}\right) X.$$` Find the eigenvalues and eigenvectors of `$A$` and construct the general solution.

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## Test for linear independence

> __Definition:__ Consider the solution function vectors `$X_1$` and `$X_2$` for a system of ODEs. The __Wronskian__ of the solutions is the determinant of the matrix `$$\left(\begin{array}{cc} x_1 & x_2 \\ y_1  & y_2\end{array}\right)$$` and is denoted by `$W(X_1,X_2).$`
<br/><br/>
> __Theorem [Abel's]:__ The solutions `$X_1$` y `$X_2$` are __linearly independents__ if `$$W(X_1,X_2)\neq 0.$$`

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## Solution for Systems of ODEs

> __Method of Eigenfunctions:__ For the system `$X'=AX.$` 
  1. Find the eigenvalues and eigenvectors of `$A.$` 
      * If the eigenvalues are real and distinct, then the general solution `$$X_1=e^{\lambda_1}v_1\mbox{ and } X_2=e^{\lambda_2}v_2.$$`
      * If the eigenvalues are real and repeated, then the general solution `$$X_1=e^{\lambda}v_1\mbox{ and } X_2=te^{\lambda}v_2.$$`
      * If the eigenvalues are complex conjugated, then the general solution `$$X_1=e^{\alpha t}\left(\cos(\beta t)v_1+\sin(\beta t)v_2\right)\mbox{ and } X_2=e^{\alpha t}\left(\cos(\beta t)v_1-\sin(\beta t)v_2\right).$$`
      
  2. Express the general solution as `$$X=c_1 X_1+c_2 X_2.$$`
  3. If there are initial conditions, use them to find the constants `$c_1$` and `$c_2.$`

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#### Example 1:

Find the general solution for `$X'=AX$` if `$A=\left(\begin{array}{cc} 1 & 2 \\ 4 & 3 \end{array}\right).$` <br/><br/>
  
  * `$A=\left(\begin{array}{cc} -4 & 2 \\ -\frac{5}{2} & 2 \end{array}\right).$` <br/><br/>

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#### Example 2:

Find the general solution for `$X'=AX$` if `$A=\left(\begin{array}{cc} 10 & -5 \\ 8 & -12 \end{array}\right).$` <br/><br/>

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#### Example 3:

Find the general solution for `$A=\left(\begin{array}{cc} 3 & -1 \\ 9 & -3 \end{array}\right).$` <br/><br/>

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