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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 12

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online