Homogeneous and Non-homogeneous Linear Systems of Equations
Explore the concepts of homogeneous and non-homogeneous linear systems of equations, equilibrium solutions, and stability analysis through linear approximations. Dive into the graphical representations and critical point stability determinations in calculus and differential equations.
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Presentation Transcript
?= 0 2 0? 1
?= 0 2 0? 1 https://www.geogebra.org/m/abg8NJzF
Chapter 7: Homogeneous linear system of equations: x = Ax Section 9.2: Non-homogeneous linear system of equations: x = Ax + b To find equilibrium solutions, set x = 0: Suppose Aw + b = 0 for some constant vector w. Then x = w is a solution to x = Ax + b : [w] = 0 = Aw + b Thus x = w is an equilibrium solution. To determine its type, translate to the origin (or graph): Let u = x w. Then x = u + w (u + w) = A(u + w) + b u + w = Au + Aw + b u = Au
?= 0 2 0? ? = 0 2 0? + 4 3 1 1 https://www.geogebra.org/m/abg8NJzF
?= 0 2 0? ? = 0 2 0? + 4 3 1 1 https://www.geogebra.org/m/abg8NJzF
?= 0 2 0? ? = 0 2 0? + 4 3 1 1 https://www.geogebra.org/m/abg8NJzF
https://www.geogebra.org/m/abg8NJzF https://www.wolframalpha.com/input/?i=streamplot+%7B+x+-+xy%2C++-y+%2B+xy%7D
9.2 Calculus 1: Linear approx. = tangent line
Locally can be approximated by a stable center Locally can be approximated by an unstable saddle Nonlinear system of differential equations can be locally approximated by linear homog DE x = Ax
We will determine stability of these critical points in 9.3 via linear approximations. Can also determine graphically (but be careful): 2 equilibrium solutions Saddle Center (or slowly expanding or contracting spiral)
https://www.wolframalpha.com/ streamplot[{y, x^2}-1] streamplot[{y, x^2-1}, {x, -2, 2}, {y, -2, 2}] https://www.geogebra.org/m/abg8NJzF Streamplot shows small trajectories, while Slope fields direction fields show small portions of tangent lines.
? ? q 2= < 0 2= > 0 Complex roots 2 positive eigenvalues 2 negative eigenvalues Two real roots p Repeated roots q < 0 implies 1 positive and 1 negative eigenvalue Two complex roots Unstable
Center is stable, but not asymptotically stable Asymptotically stable Just need one positive expo- nential for Unstable
