7.5 Homogeneous Linear Systems

What is a homogenous linear system

How to solve

• We take the Eigenvalues and Eigen vectors of it and then take the eigen vector and value

  • $x^{(1)}(t)={\vec{z}}^{(1)}{e}^{r_{1}t}and{x}^{(2)}(t)={\vec{z}}^{(2)}{e}^{r_{2}t}$ where z is the eigen vector and r is the eigen values
  • the general solution is
    • ${\vec{x}}^{\prime }(t)=c_1{\vec{z}}_1{e}^{r_{1}t}+c_2{\vec{z}}_2{e}^{r_{2}t}$

  • Remarks

    • some textbooks normalize all eigen vectors
    • the case where there is a complex eigen value is in section 7.6 complex eigen shit
    • repeated real eigen value is not in math 260

Visulaizing systems (phase plane)

• When we have 2x2 system we can visualize the solutions

  • This is called the phase plane

  • A direction field plot for ${\vec{x}}_{}^{\prime }(t)=A\vec{x}$ is obtained by plotting vectors $A\vec{x}$ at various points in the phase plane

    • A phase is the plot of solution curves (helps visualize everything)

  • We take the equilibrium solution (${\vec{x}}^{\prime }=A\vec{x},\vec{x}=0$)

    • assuming that A is a nonsingular matrix
      • with 1 solution

  • Geogebra has a nice tool