2010-03-31

# 2010-03-31 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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Unformatted text preview: MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review: Function of several variables II: Vector-valued functions 1 2. Systems of differential equations I: linear case, phase plane 1 2.1. The explicit general solution for linear systems 1 2.1.1. Observation I: Eigenvalues and eigenvectors provide solutions 2 2.1.2. Observation II: Sums and multiples of solutions are solutions 4 2.1.3. Observation III: Stability of zero 7 2.2. The phase plane 9 1. Review: Function of several variables II: Vector-valued functions • Definition • Linear approximation and the Jacobian matrix 2. Systems of differential equations I: linear case, phase plane 2.1. The explicit general solution for linear systems. Recall that • a single linear differential equation dx ( t ) dt = ax, x (0) = x has a solution x ( t ) = e at x . In particular, – If a > 0, the solution grows to infinity. – If a < 0, the solution approaches zero. • Newton’s Law of Cooling dH dt = α ( A- H ) = αA- αH dA dt = α 2 ( H- A ) = α 2 H- α 2 A where H ( t ): the object temperature at time t ; A ( t ): the ambient temperature at time t ; α : constant of proportionality; α 2 : constant of proportionality. This is an example of linear systems of differential equations. Date : 2010-03-31. 1 2 JING LI More generally, we now want to generalize the result of the single linear differential equation to the more general linear systems of differential equations: d dt x 1 ( t ) = ax 1 ( t ) + bx 2 ( t ) x 1 (0) = x 10 d dt x 2 ( t ) = cx 1 ( t ) + dx 2 ( t ) x 2 (0) = x 20 We can also write this in matrix notation (suppressing the argument t for the moment) as d dt • x 1 x 2 ‚ = • a b c d ‚• x 1 x 2 ‚ = A • x 1 x 2 ‚ , • x 1 (0) x 2 (0) ‚ = • x 10 x 20 ‚ , A = • a b c d ‚ 2.1.1. Observation I: Eigenvalues and eigenvectors provide solutions. Suppose we are looking for solutions of the form • x 1 ( t ) x 2 ( t ) ‚ = e λt • v 1 v 2 ‚ MAT 1332: CALCULUS FOR LIFE SCIENCES 3 Fact: If λ is an eigenvalue of A and v is the corresponding eigenvector, then x ( t ) = e λt v is a solution of the linear system of differential equations...
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## This note was uploaded on 03/19/2011 for the course MAT 1332 taught by Professor Munteanu during the Spring '07 term at University of Ottawa.

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2010-03-31 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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