Lecture 23

Lecture 23 - Section 5.4(Systems of Linear Differential...

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Unformatted text preview: Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont’d; What to Expect on Exam 3 April 23, 2009 2 × 2 Systems of Linear Differential Equations Today’s Session A Summary of This Session: (1) More about eigenvalues and eigenvectors of a 2 × 2 matrix. (2) 2 × 2 systems of differential equations (not necessarily linear) (3) Phase-plane method (types of nodes) 2 × 2 Systems of Linear Differential Equations Critical points, critical point set, equilibrium point(s) Example 1: Consider the system: x ′ = x- y y ′ = x + y- 2 Find the critical point(s), eigenvalues and eigenvectors, and describe completely the solution set. To find the critical points, one needs to solve, simultaneously, x ′ = 0 y ′ = 0 This means: x- y = 0 x + y- 2 = 0 So the critical point is (1,1). 2 × 2 Systems of Linear Differential Equations Example 1: Case of a Spiral To find the eigenvalues, we first put the system in the form-→ v ′ = A-→ v +-→ v Where-→ v...
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Lecture 23 - Section 5.4(Systems of Linear Differential...

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