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# 11 - MATH2860U Chapter 8 cont 1 SYSTEMS OF LINEAR FIRST...

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MATH2860U: Chapter 8 cont. .. 1 SYSTEMS OF LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS Preliminary Theory (Section 8.1 of Zill and Cullen, pg. 304) Recall: Earlier in the course, we did some work setting up systems of differential equations, but we haven’t yet talked about how to solve them. Being able to solve systems of equations is very important in applications. Application (coupled spring-mass system): Source of Image: http://aix1.uottawa.ca/~jkhoury/oscillations.htm ) ( ) ( 1 1 1 1 2 2 2 1 2 1 t F x k x x k dt x d m ) ( ) ( 2 1 2 2 2 3 2 2 2 2 t F x x k x k dt x d m Application (parallel LRC circuit): Source of Image: Modified from http://badger.physics.wisc.edu/lab/manual2/E-9.html L V dt dI RC V C I dt dV Application (double pendulum):

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MATH2860U: Chapter 8 cont. .. 2 Before we go on to solve some linear systems, let’s first introduce some terminology and understand what it means to be a solution. Definition: A first-order system of differential equations is a set of n simultaneous differential equations in n unknown functions   ) ( , ), ( ), ( 2 1 t x t x t x n . It has the form ) ..., , , , ( ) ..., , , , ( ) ..., , , , ( 2 1 2 1 2 2 2 1 1 1 n n n n n x x x t g x x x x t g x x x x t g x Definition: If each of the functions n g g g , , , 2 1 is a linear function of the dependent variables n x x x , , , 2 1 , then the system of equations is said to be linear . Thus, the normal form of n first order linear equations is: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 2 2 1 21 2 1 1 1 11 1 t f x t a x t a x t f x t a x t a x t f x t a x t a x n n nn n n n n n n Definition:
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11 - MATH2860U Chapter 8 cont 1 SYSTEMS OF LINEAR FIRST...

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