Planar Systems
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Table of Contents
- Planar Systems
#UCLA #Y1Q3 #Math33B
Planar Systems
Key Definitions
Characteristic polynomial - the determinant of the matrix, A, minus the identity, I, multiplied by $\lambda$
Planar Systems - usually homogenous linear systems with constant coefficients solved using linear algebra, specifically in 2x2 matrices below
Problem
Given a linear system:
\(\vec x'=A\vec x \quad\text{and}\quad A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}\)
With possible IVP:
\(\vec x(t_0)=\begin{bmatrix}a \\ b\end{bmatrix}\)
Steps
- Find Eigenvalues using characteristic polynomial
- Find Eigenvectors using $null(A-\lambda I_n)$
- General Solution:
\(x(t;C_1,C_2)=C_1e^{\lambda_1 t}\vec v_1 + C_2e^{\lambda_2 t}\vec v_2\)
- Plug in IVP and solve augmented matrix using general solution
General Solutions
Distinct Real Roots
Different real Eigenvalues
\(x(t)=C_1e^{\lambda_1 t}\vec v_1 + C_2e^{\lambda_2 t}\vec v_2\)
Complex Conjugate Roots
Complex Eigenvalues
\(\lambda = a + bi\)
\(\vec w = \vec v_1 + \vec v_2i\)
\(\bar\lambda = a - bi\)
\(\bar{\vec w} = \vec v_1 - \vec v_2i\)
Complex Version
\(x(t)=C_1e^{\lambda t}\vec w + C_2e^{\bar\lambda t}\bar{\vec w}\)
Real Version
\(x(t)=C_1e^{at}(\vec v_1\cos bt - \vec v_2\sin bt) + C_2e^{at}(\vec v_1\sin bt + \vec v_2\cos bt)\)
Double Real Roots
One Eigenvalue
Easy Case
2 linearly independent Eigenvectors Same as Distinct Real Roots case:
\(x(t)=C_1e^{\lambda t}\vec v_1 + C_2e^{\lambda t}\vec v_2\)
Hard Case
1 Eigenvector Find $\vec v_2$ by setting up augmented matrix:
\(\vec v_2 = \begin{bmatrix}A-\lambda I & | & \vec v_1 \end{bmatrix}\)
Then solution is given by: