Planar Systems

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Table of Contents

• Planar Systems
  • Key Definitions
  • Problem
    • ${\overrightarrow{x}}^{\prime }=A\overrightarrow{x}\text{and}A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
      • $\overrightarrow{x}(t_0)=\left[\begin{array}{c}a\\ b\end{array}\right]$
  • Steps
    • $x(t;C_1,C_2)=C_1{e}^{{\lambda }_{1}t}{\overrightarrow{v}}_1+C_2{e}^{{\lambda }_{2}t}{\overrightarrow{v}}_2$
  • General Solutions
    • Distinct Real Roots
      • $x(t)=C_1{e}^{{\lambda }_{1}t}{\overrightarrow{v}}_1+C_2{e}^{{\lambda }_{2}t}{\overrightarrow{v}}_2$
    • Complex Conjugate Roots
      • $\lambda =a+bi$
      • $\overrightarrow{w}={\overrightarrow{v}}_1+{\overrightarrow{v}}_{2}i$
      • $\overline{\lambda }=a-bi$
      • $\overline{\overrightarrow{w}}={\overrightarrow{v}}_1-{\overrightarrow{v}}_{2}i$
      • Complex Version
        • $x(t)=C_1{e}^{\lambda t}\overrightarrow{w}+C_2{e}^{\overline{\lambda }t}\overline{\overrightarrow{w}}$
      • Real Version
        • $x(t)=C_1{e}^{at}({\overrightarrow{v}}_1\cos bt-{\overrightarrow{v}}_2\sin bt)+C_2{e}^{at}({\overrightarrow{v}}_1\sin bt+{\overrightarrow{v}}_2\cos bt)$
    • Double Real Roots
      • Easy Case
        • $x(t)=C_1{e}^{\lambda t}{\overrightarrow{v}}_1+C_2{e}^{\lambda t}{\overrightarrow{v}}_2$
      • Hard Case
        • ${\overrightarrow{v}}_2=\left[\begin{array}{ccc}A-\lambda I& |& {\overrightarrow{v}}_1\end{array}\right]$
        • $x(t)=C_1{e}^{\lambda t}{\overrightarrow{v}}_1+C_2{e}^{\lambda t}({\overrightarrow{v}}_2+t{\overrightarrow{v}}_1)$

#UCLA #Y1Q3 #Math33B

Planar Systems

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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

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Problem

Given a linear system:

${\overrightarrow{x}}^{\prime }=A\overrightarrow{x}\text{and}A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

With possible IVP:

$\overrightarrow{x}(t_0)=\left[\begin{array}{c}a\\ b\end{array}\right]$

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Steps

1. Find Eigenvalues using characteristic polynomial
2. Find Eigenvectors using $null(A-\lambda I_n)$
3. General Solution:

   $x(t;C_1,C_2)=C_1{e}^{{\lambda }_{1}t}{\overrightarrow{v}}_1+C_2{e}^{{\lambda }_{2}t}{\overrightarrow{v}}_2$

4. Plug in IVP and solve augmented matrix using general solution

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General Solutions

Distinct Real Roots

Different real Eigenvalues

$x(t)=C_1{e}^{{\lambda }_{1}t}{\overrightarrow{v}}_1+C_2{e}^{{\lambda }_{2}t}{\overrightarrow{v}}_2$

Complex Conjugate Roots

Complex Eigenvalues

$\lambda =a+bi$

$\overrightarrow{w}={\overrightarrow{v}}_1+{\overrightarrow{v}}_{2}i$

$\overline{\lambda }=a-bi$

$\overline{\overrightarrow{w}}={\overrightarrow{v}}_1-{\overrightarrow{v}}_{2}i$

Complex Version

$x(t)=C_1{e}^{\lambda t}\overrightarrow{w}+C_2{e}^{\overline{\lambda }t}\overline{\overrightarrow{w}}$

Real Version

$x(t)=C_1{e}^{at}({\overrightarrow{v}}_1\cos bt-{\overrightarrow{v}}_2\sin bt)+C_2{e}^{at}({\overrightarrow{v}}_1\sin bt+{\overrightarrow{v}}_2\cos bt)$

Double Real Roots

One Eigenvalue

Easy Case

2 linearly independent Eigenvectors Same as Distinct Real Roots case:

$x(t)=C_1{e}^{\lambda t}{\overrightarrow{v}}_1+C_2{e}^{\lambda t}{\overrightarrow{v}}_2$

Hard Case

1 Eigenvector Find ${\overrightarrow{v}}_2$ by setting up augmented matrix:

${\overrightarrow{v}}_2=\left[\begin{array}{ccc}A-\lambda I& |& {\overrightarrow{v}}_1\end{array}\right]$

Then solution is given by:

$x(t)=C_1{e}^{\lambda t}{\overrightarrow{v}}_1+C_2{e}^{\lambda t}({\overrightarrow{v}}_2+t{\overrightarrow{v}}_1)$