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Lecture 22

# Lecture 22 - Section 5.4(Systems of Linear Diﬀerential...

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Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors April 21, 2009 2 × 2 Systems of Linear Differential Equations

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Today’s Session Problems to work on, but will not be posted on webassign : 9.5: 2, 4, 9, 11, 12, 18, 22, 26. Hand-in for Nov 25: 9.5 # 31. Bring your clickers on Thursday! 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 + 3 y = - 2 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 + 3 = 0 - 2 y + 2 = 0 So the critical point is (3,1). 2 × 2 Systems of Linear Differential Equations

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Example 1: Case of a Stable Node To find the eigenvalues, we first put the system in the form -→ v = A -→ v + -→ v 0 Where -→ v = parenleftbigg x y parenrightbigg , A = parenleftbigg - 1 0 0 - 2 parenrightbigg , and -→ v 0 = parenleftbigg 3 2 parenrightbigg . Note the book (in section 9.5) uses
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Lecture 22 - Section 5.4(Systems of Linear Diﬀerential...

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