Download Final Exam Review Sheet - Differential Equations | MATH 33B and more Study notes Differential Equations in PDF only on Docsity! Math 33B - Final Exam Review Sheet 1. Classification of the Equilibria of y′ = Ay where A is a 2× 2 matrix with trace T and determinant D Generic Cases: • Real eigenvalues, one positive, one negative (D < 0): Saddle • Distinct real e-values, both negative (D > 0, T 2 − 4D > 0, T < 0): Nodal Sink • Distinct real e-values, both positive (D > 0, T 2−4D > 0, T > 0): Nodal Source • Complex e-values with negative real part (T 2 − 4D < 0, T < 0): Spiral Sink • Complex e-values with positive real part (T 2 − 4D < 0, T > 0): Spiral Source Degenerate Cases: • Purely imaginary eigenvalues (D > 0, T = 0): Center • Negative e-value of multiplicity two (T 2 − 4D = 0, T < 0): Degenerate Sink (Like a nodal sink, but a solution may curve significantly as it approaches the origin, like it’s “almost” a spiral.) • Positive e-value of multiplicity two (T 2 − 4D = 0, T > 0): Degenerate Source (Like a nodal source, but a solution may curve significantly as it leaves the origin, like it’s “almost” a spiral.) • One of the eigenvalues is 0 (D = 0): No Catchy Name In this case there are infinitely many equilibria, arranged along a line in the plane. If T < 0 they are all stable. If T > 0, they are all unstable. If T = 0 they are neither. All other solutions are straight lines. 2. Qualitative Analysis of Autonomous Nonlinear Systems: { x′ = f(x, y) y′ = g(x, y) • The x-nullcline is the set of points (x, y) where f(x, y) = 0, i.e. where the arrows all point straight up or down. The y-nullcline is the set of points (x, y) where g(x, y) = 0, i.e. where the arrows all point straight left or right. • A point (x0, y0) is an equilibrium if f(x0, y0) = 0 and g(x0, y0) = 0, i.e. if it is on both nullclines. • The Jacobian of the system is the matrix J = ( ∂f ∂x ∂f ∂y ∂g ∂x ∂g ∂y ) • The linearization of the system at an equilibrium (x0, y0) is the homogeneous linear system y′ = J(x0, y0)y. • If the linearization at (x0, y0) is one of the five generic types (saddle, nodal sink, nodal source, spiral sink, spiral source), then the equilibrium at (x0, y0) is of the same type.
